NUST NET (Engineering) · Maths · Analytic geometry. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
Coordinate geometry turns geometry into algebra. Two points give you a distance, a midpoint and a gradient. The gradient m measures the steepness of a line and equals tan θ, where θ is the angle the line makes with the positive x-axis.
A line is fixed by one point and a gradient, so most questions come down to finding m and then using y − y1 = m(x − x1). For a line written as ax + by + c = 0, read the gradient straight off as −a/b.
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1. Three points are collinear when the gradients between them are equal.
Using the parallel-lines distance formula before the two equations have the same x and y coefficients. Multiply or divide one equation first so that a and b match, then compare the constants.
Answer: 3x + 2y − 7 = 0
Try each one, then open the answer.
Answer: D. Midpoint = ((3 − 7)/2, (−4 + 8)/2) = (−2, 2). (−4, 4) adds the coordinates but forgets to halve.
Answer: C. (7 − k)/(5 − 2) = 2 gives 7 − k = 6, so k = 1. Writing (k − 7)/3 = 2 gives 13.
Answer: C. tan θ = |(m1 − m2)/(1 + m1m2)| = |(3 + 2)/(1 − 6)| = 1, so θ = 45°.
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