Paper I — Mathematics

Coordinate Geometry: Straight Lines

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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High yield5 mapped NDA questions

Build the idea

Coordinate geometry turns geometry into algebra. NDA questions repeatedly combine slope, distance, line forms and angle between lines. Choose the line form that uses the information given directly: point-slope for a point and slope, two-point form for two points, and intercept form for axis intercepts.

Coordinate view

Concept

Coordinate geometry studies points, lines and curves using algebra. A straight line can be written in several forms; the slope captures its steepness and direction.

Distance, section and slope formulas are the workhorses of this topic.

Key Points & Formulas

Distance between (x₁,y₁),(x₂,y₂) = √[(x₂−x₁)²+(y₂−y₁)²].
Slope m = (y₂−y₁)/(x₂−x₁); slope-intercept form y = mx + c.
Parallel lines: equal slopes; perpendicular lines: m₁m₂ = −1.
Distance of point (x₀,y₀) from line ax+by+c=0 = |ax₀+by₀+c|/√(a²+b²).

Worked Example

Q. Find the distance of the point (3, 4) from the line 4x + 3y − 10 = 0.

Distance = |4·3 + 3·4 − 10| / √(4²+3²).

= |12 + 12 − 10| / √25 = 14/5 = 2.8 units.

Common traps

Reversing only one subtraction in the slope formula.
Forgetting the absolute value in point-to-line distance.
Treating vertical lines as having slope zero; their slope is undefined.

NDA shortcuts

Parallel: m₁=m₂. Perpendicular: m₁m₂=−1.
Use the determinant area formula to test collinearity quickly.
Keep the line in ax+by+c=0 for distances and angles.

One-screen revision sheet

Recall before you practise

m=(y₂−y₁)/(x₂−x₁)
d=|ax₀+by₀+c|/√(a²+b²)
tanθ=|(m₂−m₁)/(1+m₁m₂)|

Past-paper bank

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5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA Practice 2026

The centre of the circle $x^2+y^2-4x-6y+9=0$ is

Question 2

NDA I 2025

Let $A(3, -1)$ and $B(1, 1)$ be the end points of line segment $AB$ . Let $P$ be the middle point of the line segment $AB$ . Let $Q$ be the point situated at a distance $\sqrt{2}$ units from $P$ on the perpendicular bisector line of $AB$ . What are the possible coordinates of $Q$ ?

Question 3

NDA II 2024

If $P(2, 4)$ , $Q(8, 12)$ , $R(10, 14)$ and $S(x, y)$ are vertices of a parallelogram, then what is $(x + y)$ equal to?

Question 4

NDA II 2024

What are the coordinates of vertex $D$ ?

Question 5

NDA I 2024

ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x − y − 3 = 0 and x + y − 5 = 0. If θ is one of the equal angles, then what is cotθ equal to?

← Properties of Triangles & Heights and Distances Continue in Test Module Circle →