Coordinate

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Coordinate — Quick Facts

Cartesian Coordinate System:

• $x$-axis: horizontal axis (left to right)
• $y$-axis: vertical axis (bottom to top)
• Origin: $(0, 0)$
• A point is written as $(x, y)$

Distance Formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint Formula: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Section Formula (Internal Division): Point dividing line joining $(x_1, y_1)$ and $(x_2, y_2)$ in ratio $m:n$: $$P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$

Slope (Gradient): $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

⚡ CAT Exam Tip: A vertical line has undefined slope; a horizontal line has slope 0.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Coordinate — Study Guide

Equation of a Straight Line:

Form	Equation	Use
Slope-intercept	$y = mx + c$	Slope and y-intercept given
Point-slope	$y - y_1 = m(x - x_1)$	Slope and one point given
Two-point	$y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)$	Two points given
Intercept	$\frac{x}{a} + \frac{y}{b} = 1$	x and y intercepts given
General	$Ax + By + C = 0$	Any straight line

Worked Example:

Find equation of line through (2, 3) and (5, 7).

$m = \frac{7-3}{5-2} = \frac{4}{3}$

Using point-slope with (2, 3): $y - 3 = \frac{4}{3}(x - 2)$ $3y - 9 = 4x - 8$ $4x - 3y + 1 = 0$

Parallel and Perpendicular Lines:

• Parallel: $m_1 = m_2$
• Perpendicular: $m_1 \times m_2 = -1$

Example: Find equation of line perpendicular to $2x + 3y = 6$ passing through (4, 1).

$2x + 3y = 6 \Rightarrow m_1 = -\frac{2}{3}$ Perpendicular slope: $m_2 = \frac{3}{2}$

$y - 1 = \frac{3}{2}(x - 4)$ $2y - 2 = 3x - 12$ $3x - 2y - 10 = 0$

⚡ Common Student Mistake: Forgetting to check whether the product of slopes is $-1$ or the lines are actually perpendicular. Also, when converting between forms, ensure the sign of the slope remains correct.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Coordinate — Comprehensive Notes

Perpendicular Distance from Point to Line:

From $(x_1, y_1)$ to line $Ax + By + C = 0$: $$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

Example: Find distance from (3, 2) to line $3x + 4y + 5 = 0$.

$d = \frac{|3(3) + 4(2) + 5|}{\sqrt{9 + 16}} = \frac{|9 + 8 + 5|}{\sqrt{25}} = \frac{22}{5} = 4.4$

Area of Triangle:

Given vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$: $$\text{Area} = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$$

Example: Area of triangle with vertices (1, 2), (3, 4), (5, 1)

$A = \frac{1}{2}|1(4-1) + 3(1-2) + 5(2-4)|$ $= \frac{1}{2}|3 + 3(-1) + 5(-2)|$ $= \frac{1}{2}|3 - 3 - 10| = \frac{1}{2} \times 10 = 5$ square units

Locus Problems:

The locus of points equidistant from two fixed points $(x_1, y_1)$ and $(x_2, y_2)$ is the perpendicular bisector of the segment joining them.

Example: Find the locus of points 3 units from the x-axis.

The x-axis is the line $y = 0$. Points 3 units from this are $y = 3$ or $y = -3$.

Circle Equations:

Centre $(h, k)$, radius $r$: $(x - h)^2 + (y - k)^2 = r^2$

General form: $x^2 + y^2 + 2gx + 2fy + c = 0$ Centre: $(-g, -f)$, Radius: $\sqrt{g^2 + f^2 - c}$

Intersection of Line and Circle:

Substitute line equation into circle equation, solve quadratic. Discriminant $D$:

• $D > 0$: Two real intersections (secant)
• $D = 0$: One intersection (tangent)
• $D < 0$: No intersection

Angle Between Two Lines: $$\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$$

Worked Example:

Find the acute angle between $y = 2x + 1$ and $y = -\frac{1}{2}x + 3$.

$m_1 = 2$, $m_2 = -\frac{1}{2}$

$\tan\theta = \left|\frac{2 - (-1/2)}{1 + 2(-1/2)}\right| = \left|\frac{2.5}{0}\right|$

Since denominator is 0, $\theta = 90°$ — the lines are perpendicular.

JAMB Pattern Analysis (CAT 2015-2024):

• 2015: Distance formula
• 2017: Equation of line through two points
• 2019: Perpendicular distance
• 2021: Area of triangle
• 2023: Equation of circle
• 2024: Locus of points at a given distance

⚡ Exam Strategy: Draw the coordinate plane for any geometry-related coordinate problem. Label the given points clearly. For area, use absolute value to avoid negative results.