Mathematics · Module 3

20 lessons · conics + vectors

Geometry & Algebra — full chapter

Coordinate geometry, lines, circles, ellipses, hyperbolas and parabolas — plus vectors, complex numbers and solid geometry. Learn each idea, drill it, then prove it on previous-exam questions.

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After this chapter you can

Read quadrants and mirror points across the axes and the origin with zero sign slips.
Compute distance, midpoints, centroids and section points straight from coordinates.
Find slope from two points or $m = \tan\theta$ , and write any line in all three forms.
Test parallel/perpendicular, solve intersections, and apply point-to-line distance.
Read center and radius from circle standard form, and complete the square from general form.
Decide line–circle position with $d$ vs $r$ , and write the tangent at a point.
Find foci, vertices, axes and equations of ellipses and hyperbolas — with the correct $a,b,c$ relation each time.
Find focus, directrix and focal radius of parabolas opening in any direction.
Build vectors tip-minus-tail, use the dot product for angles and perpendicularity, divide complex numbers via conjugates, and compute solid volumes.

How to study this chapter

Read the concept in simple words.
Write down the exact formula.
Work one easy example by hand.
Name the trap for this type out loud.
Solve Basic → Intermediate → Advanced.
Finish with mixed + previous-exam questions.

Lessons

Lesson 1 of 20

Quadrants, Axes and Symmetry Points

Learn the idea

Quadrant I is

(+, +)

, II is

(-, +)

, III is

(-, -)

, IV is

(+, -)

. Check the sign of

x

first, then

y

Mirror across the x-axis flips

y

only:

(a, b) \to (a, -b)

. Across the y-axis flips

x

only; across the origin flips both.

A point with

|y| = k

and fixed

x

splits into two points

(x, k)

and

(x, -k)

Key rules to memorize

QI $(+,+)$ ; QII $(-,+)$ ; QIII $(-,-)$ ; QIV $(+,-)$
x-axis mirror: $(a,b) \to (a,-b)$ ; y-axis mirror: $(a,b) \to (-a,b)$
Origin mirror: $(a,b) \to (-a,-b)$
$|y| = k$ with fixed $x$ gives two points: $(x, k)$ and $(x, -k)$
$3 - \pi \approx -0.14$ : compare against $0$ , never guess the sign

Worked example

$P(1,2)$ ; $Q$ symmetric about the x-axis: $Q =$ ?

x-axis symmetry flips $y$ only.
$(1,2) \to (1,-2)$ .

$(1,-2)$

Classic trap

Mirroring

(1,2)

across the x-axis as

(2,1)

: swapping is for

y = x

, not for axis mirrors.

Lesson 2 of 20

Distance Formula

Learn the idea

Distance is Pythagoras on coordinate differences:

d = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}

Subtract first, then square — squaring before subtracting is the classic slip.

Recognize triples (3-4-5, 5-12-13, 8-15-17) to finish in seconds.

Key rules to memorize

$d = \sqrt{(\Delta x)^{2} + (\Delta y)^{2}}$
Subtract coordinates first, square second
3-4-5 scaled: $\sqrt{64+1} = \sqrt{65}$ is NOT a triple — compute it
$|AB| = 5$ with one fixed point gives $x^{2} + 16 = 25$ : both signs work

Worked example

Distance $P(-1,2)$ to $Q(3,1)$ .

$\Delta x = 4$ , $\Delta y = -1$ .
$d = \sqrt{16+1}$ .

$\sqrt{17}$

Classic trap

Writing

\sqrt{x_{2}^{2} - x_{1}^{2}}

instead of

\sqrt{(x_{2}-x_{1})^{2}}

: subtract, then square.

Lesson 3 of 20

Midpoint and Section Formula

Learn the idea

Midpoint averages each coordinate:

(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})

As vectors: the midpoint position is

\frac{a+b}{2}

— tip minus tail builds the segment, average finds its middle.

Centroid averages all three vertices; section formula weights by the opposite ratio.

Key rules to memorize

Midpoint: average each coordinate
Vector form: midpoint of $AB$ is $\frac{a+b}{2}$
Centroid: $(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3})$
Section $\lambda{:}\mu$ : $(\frac{\mu x_{1}+\lambda x_{2}}{\lambda+\mu}, \ldots)$
$|AB| = 5$ with $A(0,3)$ , $B(x,-1)$ : $x^{2} = 9$ , so $x = \pm 3$

Worked example

Midpoint of $(1,2)$ and $(5,6)$ .

Average $x$ : $\frac{1+5}{2} = 3$ .
Average $y$ : $\frac{2+6}{2} = 4$ .

$(3,4)$

Classic trap

Averaging

x

with

y

(e.g.

(\frac{1+2}{2}

for

(1,2)

(5,6)

) — average each coordinate with its own kind.

Lesson 4 of 20

Slope and Inclination

Learn the idea

Slope measures direction:

m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}

, and also

m = \tan\theta

for inclination

\theta

Memorize the tan ladder:

30° \to \frac{\sqrt{3}}{3}

45° \to 1

60° \to \sqrt{3}

90°

undefined,

> 90°

negative.

Same x-coordinate twice means a vertical line: undefined slope, handle separately.

Key rules to memorize

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \tan\theta$
$0° \to 0$ ; $30° \to \frac{\sqrt{3}}{3}$ ; $45° \to 1$ ; $60° \to \sqrt{3}$
$135° \to -1$ ; $120° \to -\sqrt{3}$ ; $90°$ undefined
Vertical line $x = c$ : undefined slope — never divide by zero
Sign of slope = direction of the linear function

Worked example

Slope through $A(-2,3)$ , $B(3,1)$ .

$m = \frac{1-3}{3-(-2)}$ .
Simplify $-\frac{2}{5}$ .

$-\frac{2}{5}$

Classic trap

Computing

\frac{1-3}{3+2}

\frac{2}{5}

: keep the numerator sign — it is

-\frac{2}{5}

Lesson 5 of 20

Line Forms and Intercepts

Learn the idea

Three forms, one line:

y = mx+b

y-y_{1} = m(x-x_{1})

Ax+By+C = 0

. Pick by data: slope+point → point-slope; two points → slope first.

From general form: slope

m = -\frac{A}{B}

, x-intercept

-\frac{C}{A}

, y-intercept

-\frac{C}{B}

Horizontal is

y = c

(

m = 0

); vertical is

x = c

(undefined slope).

Key rules to memorize

Point-slope: $y-y_{1} = m(x-x_{1})$
General: $Ax+By+C = 0$ ; slope $m = -\frac{A}{B}$ , intercepts $-\frac{C}{A}$ , $-\frac{C}{B}$
Point-slope with slope $-3$ through $(1,2)$ : $3x+y-5 = 0$
Inclination $45°$ through $(0,2)$ : $m = 1$ , so $y = x+2$
x-intercept sets $y = 0$ ; y-intercept sets $x = 0$

Worked example

Slope $-3$ through $(1,2)$ : equation?

Point-slope: $y - 2 = -3(x-1)$ .
Expand: $y = -3x+5$ , i.e. $3x+y-5 = 0$ .

$3x+y-5 = 0$

Classic trap

Expanding

y-2 = -3(x-1)

3x+y+1 = 0

-3 \times -1 = +3

, so

3x+y-5 = 0

Lesson 6 of 20

Parallel and Perpendicular Lines

Learn the idea

Parallel means equal slopes:

m_{1} = m_{2}

(vertical lines are parallel to each other).

Perpendicular (non-vertical) means

m_{1}m_{2} = -1

: flip and negate.

In general form

Ax+By+C = 0

, perpendicular swaps coefficients: direction

(A,B)

turns into

(-B,A)

Key rules to memorize

Parallel: $m_{1} = m_{2}$
Perpendicular: $m_{1}m_{2} = -1$ (non-vertical)
Given slope $-1$ : perpendicular slope is $1$
Given slope $2$ : perpendicular slope is $-\frac{1}{2}$
General-form perpendicular to $x+2y+4 = 0$ : direction $(1,2) \to (-2,1)$ , i.e. slope $2$

Worked example

Which line is perpendicular to $x+y+3 = 0$ ?

Given slope $-1$ .
Perpendicular slope $1$ : $y = x+3$ .

$y = x+3$

Classic trap

Using

m_{1}m_{2} = -1

on a vertical/horizontal pair: one slope is undefined, so construct directly instead.

Lesson 7 of 20

Intersections and Point-to-Line Distance

Learn the idea

Intersection solves both equations at once: substitute when one line is in slope form, eliminate for two general forms.

Always verify in both lines — a slip survives a single check.

Point-to-line distance needs general form first:

\frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}

Key rules to memorize

Substitute (slope form) or eliminate (two general forms)
Verify the point in BOTH lines
Distance: $\frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$
Parallel-line distance: $\frac{|C_{1}-C_{2}|}{\sqrt{A^{2}+B^{2}}}$ after normalizing $A,B$
Perpendicular condition in coefficients: $a_{1}a_{2}+b_{1}b_{2} = 0$

Worked example

Intersection of $y = 2x+1$ and $x+y+1 = 0$ .

Substitute: $x+2x+1+1 = 0 \Rightarrow x = -\frac{2}{3}$ .
Then $y = -\frac{1}{3}$ ; check both.

$(-\frac{2}{3},-\frac{1}{3})$

Classic trap

Forgetting the absolute value in the distance formula, or using unnormalized

(A,B)

when comparing two parallel lines.

Lesson 8 of 20

Circle Standard Form

Learn the idea

Standard form shows everything:

(x-a)^{2}+(y-b)^{2} = r^{2}

has center

(a,b)

and radius

r

Signs flip when reading:

(x+3)^{2}

means center

x = -3

The right side is

r^{2}

, not

r

: take the square root for the radius.

Key rules to memorize

Center $(a,b)$ , radius $r$ from $(x-a)^{2}+(y-b)^{2} = r^{2}$
$(x+3)^{2} \Rightarrow$ center $x = -3$
Right side $16 \Rightarrow r = 4$ ; right side $4 \Rightarrow r = 2$
Radius from a point on the circle: distance from center (e.g. vertical gap $2$ )
Diameter endpoints give center (midpoint) and radius (half the distance)

Worked example

Center $(-3,2)$ , radius $4$ : equation?

$(x+3)^{2} + (y-2)^{2} = 4^{2}$ .
Simplify the right side.

$(x+3)^{2}+(y-2)^{2} = 16$

Classic trap

Reading center

(-3,2)

from

(x+3)^{2}+(y-2)^{2} = 16

(3,-2)

: signs flip.

Lesson 9 of 20

Circle General Form

Learn the idea

General form

x^{2}+y^{2}+Dx+Ey+F = 0

hides center and radius: complete the square in

x

and

y

independently.

Shortcut: center

(-\frac{D}{2},-\frac{E}{2})

, radius

r = \frac{\sqrt{D^{2}+E^{2}-4F}}{2}

Reality check first:

D^{2}+E^{2}-4F > 0

, else it is a point or empty.

Key rules to memorize

Complete the square in $x$ and $y$ separately
Center $(-\frac{D}{2},-\frac{E}{2})$ ; $r = \frac{\sqrt{D^{2}+E^{2}-4F}}{2}$
Reality: $> 0$ circle, $= 0$ point, $< 0$ empty
Example: $x^{2}+y^{2}-4x-3 = 0 \Rightarrow (x-2)^{2}+y^{2} = 7$
Example: $x^{2}+y^{2}+2x-2y = 0 \Rightarrow r = \sqrt{2}$

Worked example

Center and radius of $x^{2}+y^{2}-4x-3 = 0$ .

Complete squares: $(x-2)^{2} + y^{2} = 7$ .
Read center and radius.

Center $(2,0)$ , $r = \sqrt{7}$

Classic trap

Halving only the inside of the root in the radius formula — the

/2

applies to the whole root.

Lesson 10 of 20

Line–Circle Position and Tangents

Learn the idea

Position is one comparison: center-to-line distance

d

vs radius

r

: separate (

d > r

), tangent (

d = r

), secant (

d < r

Only solve the full substitution when the actual points are asked; for position, compare

d

with

r

Radius to a point of tangency is perpendicular to the tangent line.

Key rules to memorize

$d > r$ separate; $d = r$ tangent; $d < r$ secant (two points)
Tangent $\perp$ radius at the point of tangency
Tangent to $x^{2}+y^{2} = r^{2}$ at $(x_{0},y_{0})$ : $xx_{0}+yy_{0} = r^{2}$
Center $(0,0)$ , $r = 1$ , line $y = x$ : $d = 0 < 1$ so two points
Through $A(2,2)$ , center $(2,0)$ : radius is the vertical gap $2$

Worked example

Does $y = x$ meet $x^{2}+y^{2} = 1$ , and how often?

Center $(0,0)$ , $r = 1$ ; distance from center to $y = x$ is $0$ .
$0 < 1$ : secant.

Yes — two intersection points

Classic trap

Full substitution when only the count (0/1/2 points) is asked — compare

d

with

r

instead.

Lesson 11 of 20

Ellipses: Foci, Vertices, Axes

Learn the idea

Ellipse = constant SUM of distances to two foci (sum

= 2a

The larger denominator names the major axis; foci and vertices lie on it.

Bridge with

a^{2} = b^{2}+c^{2}

— minus, unlike the hyperbola.

Key rules to memorize

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ ( $a > b$ ): foci $(\pm c,0)$ , vertices $(\pm a,0)$
$a^{2} = b^{2}+c^{2}$
Focal radii sum: $PF_{1}+PF_{2} = 2a$
Vertical major axis: foci $(0,\pm c)$ , vertices $(0,\pm a)$
Eccentricity $e = \frac{c}{a}$ , $0 < e < 1$ : near 0 is near-circular

Worked example

Foci of $\frac{x^{2}}{25}+\frac{y^{2}}{9} = 1$ .

$a^{2} = 25$ , $b^{2} = 9$ : $c^{2} = 16$ .
Major axis on x: foci $(\pm 4,0)$ .

$(4,0)$ and $(-4,0)$

Classic trap

Putting foci on the minor axis — the LARGER denominator names the major axis.

Lesson 12 of 20

Ellipse Equations From Data

Learn the idea

Translate data into

a

b

c

: vertices give

a

, foci give

c

, minor endpoints give

b

Missing one? Use

a^{2} = b^{2}+c^{2}

e = \frac{c}{a}

, or focal length

2c

Place the major variable correctly: y-major means

\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1

Key rules to memorize

Vertices $\Rightarrow a$ ; foci $\Rightarrow c$ ; minor endpoints $\Rightarrow b$
$e = \frac{c}{a}$ : $a = \frac{c}{e}$ ; focal length $2c$
Foci on y-axis, $c = 2$ , $e = \frac{2}{3}$ : $a = 3$ , $b^{2} = 5$
Same foci as $\frac{x^{2}}{4}+\frac{y^{2}}{9} = 1$ ( $c^{2} = 5$ ), minor $4\sqrt{5}$ : $b^{2} = 20$ , $a^{2} = 25$
Ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2} = 1$ : focal sum $2a = 4$

Worked example

Foci on y-axis, focal length $4$ , $e = \frac{2}{3}$ : equation?

$c = 2$ ; $a = \frac{c}{e} = 3$ ; $b^{2} = 9-4 = 5$ .
Major on y: $\frac{x^{2}}{5}+\frac{y^{2}}{9} = 1$ .

$\frac{x^{2}}{5}+\frac{y^{2}}{9} = 1$

Classic trap

Writing

\frac{x^{2}}{9}+\frac{y^{2}}{5} = 1

for y-major data: the major denominator

9

belongs under

y

Lesson 13 of 20

Hyperbolas: Foci, Vertices, Asymptotes

Learn the idea

Hyperbola = constant DIFFERENCE of distances to two foci (difference

= 2a

The positive term names the transverse axis; bridge with

c^{2} = a^{2}+b^{2}

— plus, unlike the ellipse.

Asymptotes

y = \pm(\frac{b}{a})x

(horizontal form): sketch the

\pm a

\pm b

box first.

Key rules to memorize

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1$ : foci $(\pm c,0)$ , vertices $(\pm a,0)$
$c^{2} = a^{2}+b^{2}$
Asymptotes: $y = \pm(\frac{b}{a})x$
$|PF_{1}-PF_{2}| = 2a$ ; $e = \frac{c}{a} > 1$
Example: $x^{2}-y^{2} = 1$ : $a^{2} = b^{2} = 1$ , foci $(\pm\sqrt{2},0)$

Worked example

Foci of $x^{2}-y^{2} = 1$ .

$a^{2} = b^{2} = 1$ , so $c^{2} = 2$ .
Foci $(\pm\sqrt{2},0)$ .

$(\pm\sqrt{2},0)$

Classic trap

Using the ellipse relation

a^{2} = b^{2}+c^{2}

: hyperbolas ADD (

c^{2} = a^{2}+b^{2}

Lesson 14 of 20

Hyperbola Equations and Eccentricity

Learn the idea

Vertices give

a

; asymptotes

y = \pm(\frac{b}{a})x

give

b

;

e = \frac{c}{a}

with

c^{2} = a^{2}+b^{2}

closes the loop.

Focal distance is

2c

: halve before using

a

b

relations.

Vertical form

\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1

: foci

(0,\pm c)

, asymptotes

y = \pm(\frac{a}{b})x

Key rules to memorize

Vertices $(\pm 2,0)$ , asymptotes $y = \pm x$ : $a = b = 2$
$e = 2$ : $c = 2a$ , so $4a^{2} = a^{2}+b^{2}$
$\frac{x^{2}}{64}-\frac{y^{2}}{16} = 1$ : $c^{2} = 80$ , focal distance $8\sqrt{5}$
Focal distance $2c$ vs focal length $c$ : halve first
Vertical form swaps asymptote slope to $\frac{a}{b}$

Worked example

Hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{3} = 1$ , $e = 2$ : $a =$ ?

$c = 2a$ ; $c^{2} = a^{2}+3$ .
$4a^{2} = a^{2}+3 \Rightarrow a^{2} = 1$ .

$a = 1$

Classic trap

Reporting

c

when asked for focal distance

2c

(e.g.

4\sqrt{5}

instead of

8\sqrt{5}

Lesson 15 of 20

Parabolas Opening Left–Right

Learn the idea

Form

y^{2} = 2px

: opens right if

p > 0

, left if

p < 0

Focus

(\frac{p}{2},0)

, directrix

x = -\frac{p}{2}

: symmetric about the vertex, distance

\frac{|p|}{2}

each way.

Focal radius: a point's distance to focus equals

x

-coordinate shifted by

\frac{p}{2}

Key rules to memorize

$y^{2} = 2px$ : focus $(\frac{p}{2},0)$ , directrix $x = -\frac{p}{2}$
$y^{2} = 4x$ : $2p = 4$ , focus $(1,0)$ — only ONE focus
Focal radius at $x$ : $x + \frac{p}{2}$ (here $4+1 = 5$ )
$y^{2} = -x$ : $2p = -1$ , directrix $x = \frac{1}{4}$
Negative $p$ flips opening to the left

Worked example

Focus of $y^{2} = 4x$ ; point with $x = 4$ : $|PF| =$ ?

Focus $(1,0)$ : $\frac{p}{2} = 1$ .
Focal radius $4+1$ .

$5$

Classic trap

Two foci for a parabola (e.g.

(\pm 1,0)

): parabolas have exactly one focus.

Lesson 16 of 20

Parabolas Opening Up–Down and Vertex Form

Learn the idea

Form

x^{2} = 2py

: opens up if

p > 0

, down if

p < 0

; focus

(0,\frac{p}{2})

, directrix

y = -\frac{p}{2}

Vertex form

y = a(x-h)^{2}+k

: vertex

(h,k)

, sign of

a

gives direction.

Through a point: substitute to solve for

p

a

Key rules to memorize

$x^{2} = 2py$ : focus $(0,\frac{p}{2})$ , directrix $y = -\frac{p}{2}$
Opens up through $(2,1)$ : $4 = 2p \Rightarrow p = 2$ , equation $x^{2} = 4y$
Vertex form vertex $(h,k)$ ; $a > 0$ opens up
Focal length $= \frac{|p|}{2}$
Axis of $y^{2} = 2px$ is the x-axis; of $x^{2} = 2py$ the y-axis

Worked example

Opens up, vertex at origin, through $(2,1)$ : equation?

Form $x^{2} = 2py$ ; substitute: $4 = 2p$ .
$p = 2$ : $x^{2} = 4y$ .

$x^{2} = 4y$

Classic trap

Squaring the wrong variable for the stated direction (

y^{2}

for up/down).

Lesson 17 of 20

Vectors: Coordinates, Magnitude, Midpoints

Learn the idea

Build vectors tip-minus-tail:

AB = b-a

; add and scale component-wise.

Magnitude is Pythagoras:

|a| = \sqrt{x^{2}+y^{2}}

; unit vector is

\frac{a}{|a|}

Midpoint position is

\frac{a+b}{2}

; collinear points differ by a scalar multiple.

Key rules to memorize

$AB = b-a$ (tip minus tail)
$|a| = \sqrt{x^{2}+y^{2}}$ ; Magnet unit: $\frac{a}{|a|}$
Midpoint: $\frac{a+b}{2}$ — true midpoint is $\frac{OA+OB}{2}$ , not a difference
Collinear: $BD$ continues $BC$ when $BC+CD$ is a multiple (e.g. $AB$ )
Square $ABCD$ , $E$ midpoint of $CD$ : $BE = BC+CE = b-\frac{a}{2}$

Worked example

Square $ABCD$ , $E$ midpoint of $CD$ , $AB = a$ , $AD = b$ : $BE =$ ?

Walk: $BE = BC+CE$ .
Substitute $BC = b$ , $CE = -\frac{a}{2}$ .

$b-\frac12 a$

Classic trap

Reversing to

a-b

for

AB

, or answering

OB-OA

for a midpoint (that is

AB

, not the midpoint position).

Lesson 18 of 20

Dot Product, Angles, Parallelism

Learn the idea

Two faces:

a\cdot b = x_{1}x_{2}+y_{1}y_{2} = |a||b|\cos\theta

Perpendicular (nonzero) vectors have zero dot product; parallel vectors have proportional components.

Set coordinate form equal to geometric form to find angles.

Key rules to memorize

$a\cdot b = x_{1}x_{2}+y_{1}y_{2} = |a||b|\cos\theta$
$a \perp b \iff a\cdot b = 0$ (nonzero)
$a \parallel b \iff x_{1}y_{2} = x_{2}y_{1}$ (2D)
$\cos\theta = \frac{a\cdot b}{|a||b|}$
Perpendicular slopes: $a_{1}a_{2}+b_{1}b_{2} = 0$ (e.g. $a = 0, -1$ )

Worked example

Are $a = (1,2)$ and $b = (-2,1)$ perpendicular?

Compute $a\cdot b = 1\cdot(-2)+2\cdot1$ .
Equals $0$ .

Yes — dot product is 0

Classic trap

Concluding perpendicular from a single zero component rather than a zero dot product.

Lesson 19 of 20

Complex Numbers

Learn the idea

Write

z = a+bi

; add/subtract component-wise, multiply with FOIL and

i^{2} = -1

Divide by the conjugate:

(a+bi)(a-bi) = a^{2}+b^{2}

real-izes the denominator.

Modulus

|z| = \sqrt{a^{2}+b^{2}}

;

\Delta < 0

quadratics give conjugate pairs; reduce

i

-powers mod 4.

Key rules to memorize

$i^{2} = -1$ ; FOIL then replace
Divide via conjugate: denominator becomes $a^{2}+b^{2}$
$|a+bi| = \sqrt{a^{2}+b^{2}}\]; \(|z|^{2} = z\bar{z}$
$z^{3} = 1$ , $z \ne 1$ : $1+z+z^{2} = 0$ (1000 terms = 333 blocks + 1)
$z = a+ai$ , $|z|^{2} = 2a^{2}$ ; $\Delta < 0 \Rightarrow$ conjugate roots
Vieta pair: $x^{2}+mx+c = 0$ with roots $z,\bar{z}$ gives sum $2\Re z = -m$ and product $|z|^{2} = c$

Worked example

$(z-2i)(2-i) = 5$ : $z =$ ?

Isolate: $z-2i = \frac{5}{2-i} = 2+i$ .
Add $2i$ .

$z = 2+3i$

Classic trap

Leaving

i^{2}

un-replaced, or rationalizing with the wrong (same-sign) conjugate.

Lesson 20 of 20

Solid Geometry

Learn the idea

Classify first: two lines are parallel, intersecting, or skew; a line is parallel to, in, or cutting a plane.

Volume ladder: prisms/cylinders

V = Bh

, pyramids/cones

V = \frac{Bh}{3}

, sphere

V = \frac{4\pi r^{3}}{3}

Cut axial cross-sections to reduce 3D items to 2D Pythagoras/area reads. No previous-exam items in the bank yet — this lesson + drills get ahead of the paper.

Key rules to memorize

Parallel / intersecting / skew lines; line–plane positions
Prism/cylinder $V = Bh$ ; pyramid/cone $V = \frac{Bh}{3}$ ; sphere $V = \frac{4\pi r^{3}}{3}$ , $S = 4\pi r^{2}$
Cone slant height $l = \sqrt{r^{2}+h^{2}}\]; lateral area \(\pi rl$
Space diagonal of a box: $\sqrt{a^{2}+b^{2}+c^{2}}$
Lateral vs total surface: sides only vs plus bases

Worked example

Cone $r = 3$ , $h = 4$ : volume?

Slant check not needed for volume: $V = \frac{\pi r^{2}h}{3}$ .
Compute $\pi \cdot 9 \cdot \frac{4}{3}$ .

$12\pi$

Classic trap

Forgetting the

/3

on pyramid/cone volumes, or using height

h

where slant height

l

is needed.

Master formula sheet

Coordinates

QI $(+,+)$ ; QII $(-,+)$ ; QIII $(-,-)$ ; QIV $(+,-)$
x-mirror $(a,b)\to(a,-b)$ ; y-mirror $(a,b)\to(-a,b)$ ; origin $(a,b)\to(-a,-b)$
$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$ ; subtract first, square second
Midpoint $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$ ; centroid averages all three vertices
Section $\lambda{:}\mu$ : $(\frac{\mu x_{1}+\lambda x_{2}}{\lambda+\mu}, \frac{\mu y_{1}+\lambda y_{2}}{\lambda+\mu})$

Lines

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \tan\theta$ ; $30°\to\frac{\sqrt{3}}{3}$ , $45°\to1$ , $60°\to\sqrt{3}$ , $90°$ undefined
Point-slope $y-y_{1} = m(x-x_{1})$ ; slope form $y = mx+b$ ; general $Ax+By+C = 0$
General form: slope $m = -\frac{A}{B}$ , x-intercept $-\frac{C}{A}$ , y-intercept $-\frac{C}{B}$
Parallel $m_{1} = m_{2}$ ; perpendicular $m_{1}m_{2} = -1$ (non-vertical)
Point-to-line: $\frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}\]}; parallel lines: \frac{\(|C_{1}-C_{2}|}{\sqrt{A^{2}+B^{2}}}$

Circles

Standard $(x-a)^{2}+(y-b)^{2} = r^{2}$ : center $(a,b)$ , radius $r$
General: center $(-\frac{D}{2},-\frac{E}{2})$ , $r = \frac{\sqrt{D^{2}+E^{2}-4F}}{2}$ ; need $D^{2}+E^{2}-4F > 0$
$d > r$ separate; $d = r$ tangent; $d < r$ secant (two points)
Tangent to $x^{2}+y^{2} = r^{2}$ at $(x_{0},y_{0})$ : $xx_{0}+yy_{0} = r^{2}$

Conics

Ellipse: larger denominator is the major axis; $a^{2} = b^{2}+c^{2}$ (minus); $PF_{1}+PF_{2} = 2a$ ; $e = \frac{c}{a} < 1$
Hyperbola: positive term is transverse; $c^{2} = a^{2}+b^{2}$ (plus); $|PF_{1}-PF_{2}| = 2a$ ; asymptotes $y = \pm(\frac{b}{a})x$ ; $e > 1$
Parabola $y^{2} = 2px$ : focus $(\frac{p}{2},0)$ , directrix $x = -\frac{p}{2}$ ; $x^{2} = 2py$ : focus $(0,\frac{p}{2})$ , directrix $y = -\frac{p}{2}$
Focal radius on $y^{2} = 2px$ : $x + \frac{p}{2}$ ; focal distance of ellipse/hyperbola is $2c$

Vectors · Complex · Solid

$AB = b-a$ (tip minus tail); midpoint position $\frac{a+b}{2}$
$|a| = \sqrt{x^{2}+y^{2}}\]; unit vector \(a/|a|$
$a\cdot b = x_{1}x_{2}+y_{1}y_{2} = |a||b|\cos\theta$ ; $a\perp b \iff a\cdot b = 0$ ; $a\parallel b \iff x_{1}y_{2} = x_{2}y_{1}$
$i^{2} = -1$ ; divide via conjugate: $(a+bi)(a-bi) = a^{2}+b^{2}$ ; $|a+bi| = \sqrt{a^{2}+b^{2}}$
Prism/cylinder $V = Bh$ ; pyramid/cone $V = \frac{Bh}{3}$ ; sphere $V = \frac{4\pi r^{3}}{3}$ , $S = 4\pi r^{2}$

Traps & recognition

10 classic traps

Reading center $(3,-2)$ from $(x+3)^{2}+(y-2)^{2} = 16$ : signs flip — it is $(-3,2)$ , radius $4$ not $16$ .
Squaring before subtracting in distance: always $(x_{2}-x_{1})^{2}$ , never $x_{2}^{2}-x_{1}^{2}$ .
Using the ellipse relation on a hyperbola: ellipses subtract $(a^{2} = b^{2}+c^{2})$ , hyperbolas add $(c^{2} = a^{2}+b^{2})$ .
Putting foci on the minor axis: the LARGER denominator names the major axis — foci follow it.
Reporting $c$ when asked for focal distance $2c$ (halve/double check every time).
Giving a parabola two foci: exactly one focus exists (e.g. $y^{2} = 4x$ has only $(1,0)$ ).
Mirroring $(1,2)$ across the x-axis as $(2,1)$ : swapping is for $y = x$ ; axis mirrors flip one sign.
Applying $m_{1}m_{2} = -1$ to a vertical/horizontal pair: one slope is undefined — construct directly.
Forgetting $/3$ on pyramid/cone volumes, or using height where slant height $l$ belongs.
Leaving $i^{2}$ un-replaced, or answering $OB-OA$ for a midpoint (that is $AB$ ; midpoint is $\frac{OA+OB}{2}$ ).

Fast problem recognition

Quadrant / symmetry point
Check $x$ sign, then $y$ sign; mirrors flip one coordinate: $(a,b)\to(a,-b)$ on the x-axis.
Two points + distance
Subtract first, square second: \(\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}\]; watch for 3-4-5, 5-12-13 triples.
Slope or inclination
Two points $\to \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ ; angle $\to \tan\theta$ ; vertical $\to$ undefined, handle apart.
Parallel / perpendicular
Equal slopes vs $m_{1}m_{2} = -1$ ; in general form swap $(A,B)\to(-B,A)$ for perpendicular.
Circle equation
Standard form: flip signs, root the right side; general form: center $(-\frac{D}{2},-\frac{E}{2})$ , check $D^{2}+E^{2}-4F > 0$ .
Line meets circle — how many points?
Compare $d$ vs $r$ (0/1/2 points); full substitution only when the points themselves are asked.
Ellipse vs hyperbola
Plus–plus with minus relation ( $a^{2} = b^{2}+c^{2}$ ) vs minus with plus relation ( $c^{2} = a^{2}+b^{2}$ ); foci sit on the major/transverse axis.
Parabola data
Squared variable names the axis; $2p$ coefficient gives $p$ ; focus/directrix sit $\frac{|p|}{2}$ each way; one focus only.

Practice — 1000 drill problems

100 core drills plus 900 lesson-tagged extra drills, basic to exam-pattern and new-type variants, each with the thinking shown. Master these and the previous-exam set below, and the chapter is yours. Tagged by lesson so you can drill exactly what you missed.

1000 practice problems · 0/0 correct

Practice · Q1geom-quadrants
$(-2,-5)$ lies in
Practice · Q2geom-quadrants
$(4,-3)$ mirrored about the x-axis:
Practice · Q3geom-quadrants
$(-1,6)$ mirrored about the y-axis:
Practice · Q4geom-quadrants
$(2,5)$ mirrored about the origin:
Practice · Q5geom-quadrants
$M$ has $x = 3$ , distance $2$ from the x-axis: $M =$
Practice · Q6geom-distance
Distance $(0,0)$ to $(3,4)$ :
Practice · Q7geom-distance
Distance $(1,1)$ to $(4,5)$ :
Practice · Q8geom-distance
Distance $P(-1,2)$ to $Q(3,1)$ :

Previous CSCA exam questions

Real Geometry & Algebra questions from the Dec 2025, Jan 2026 and Apr 2026 papers — solved with the thinking shown. Answers solved by the app; items with source errors are marked and still teach the correct math.

48 exam questions · 0/0 correct

Dec 2025 · Q6analytic-geometry
The point $(-1, 3-\pi)$ lies in which quadrant?
Dec 2025 · Q11analytic-geometry
Which point lies in the third quadrant?
Dec 2025 · Q14lines
The inclination angle of $y = \sqrt{3}x + 10$ is
Dec 2025 · Q18circles
Circle center $(-3,2)$ , radius 4: equation is
Dec 2025 · Q21circles
Circle $x^2+y^2+2x-2y = 0$ : radius is
Dec 2025 · Q25analytic-geometry
Distance $P(-1,2)$ to $Q(3,1)$ is
Dec 2025 · Q28lines
Intersection of $y = 2x+1$ and $x+y+1 = 0$ is
Dec 2025 · Q29hyperbolas
Foci of $x^2 - y^2 = 1$ are

Related — Geometry skills tested elsewhere (3)

Strictly increasing $y = 2x+1$ : positive slope means the line rises everywhere — sign of slope = direction (lesson 4).
Decreasing $y = -x+5$ : negative slope means the line falls everywhere (lesson 4).
$a_n = 2n-1$ : linear-in- $n$ form with constant step $2$ reads like slope — monotonic like a linear function (lessons 4–5).

3 exam questions · 0/0 correct

Dec 2025 · Q10monotonicity
Which function is strictly increasing?
Apr 2026 · Q11monotonicity
Decreasing on $(-\infty,+\infty)$ :
Apr 2026 · Q17arithmetic-sequences
$a_n = 2n-1$ : correct is

Exam checklist

0/15 checked

Analytic basics

Lines

Circles & conics

Vectors · complex · solid

One-minute memory sheet

Mirror flips one sign: $(a,b)\to(a,-b)$ on the x-axis.
Distance: subtract, then square: $\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}$ .
Slope is $\tan\theta$ : $45°\to1$ , $60°\to\sqrt{3}$ , $90°$ undefined.
Circle: flip signs for center, root the right side for $r$ .
Ellipse MINUS, hyperbola PLUS: $a^{2} = b^{2}+c^{2}$ vs $c^{2} = a^{2}+b^{2}$ .
Parabola has ONE focus; $2p$ gives $p$ .
$AB = b-a$ ; midpoint is $\frac{a+b}{2}$ ; $a\cdot b = 0$ means perpendicular.
Pyramid/cone volume carries $/3$ ; $i^{2} = -1$ .

Related guides

30-day study plan → fit this chapter into a week-by-week system
75-second pacing rule → hold 75s/question once the content clicks
Top 10 mistakes → the error patterns that cost the most points

Last reviewed Sep 2026 · GetCSCA editors

Timed practice →Back to syllabus →