Board 2026 Class 12 Maths Q20 Solution: Perpendicular lines x = py + q, z = ry + s

The question

Select the correct answer from the codes (A), (B), (C) and (D) as given below.

(A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).

(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

(C) Assertion (A) is true and Reason (R) is false.

(D) Assertion (A) is false and Reason (R) is true.

Assertion (A) : Lines given by $x = py + q$ ⁠, $z = ry + s$ and $x = p'y + q'$ ⁠, $z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$ ⁠.

Reason (R) : Two lines $\vec r = \vec a_1 + \lambda\vec b_1$ and $\vec r = \vec a_2 + \mu\vec b_2$ are perpendicular to each other if $\vec b_1 \cdot \vec b_2 = 0$ ⁠.

Exam answer

Put $y = t$ in the first line: $x = pt + q$ ⁠, $y = t$ ⁠, $z = rt + s$ ⁠. So its direction ratios are $p, 1, r$ ⁠.

Why?

With $y = t$ ⁠, every point of the line is

(q, 0, s) + t(p, 1, r)

As $t$ changes, the point moves in the direction $p, 1, r$ ⁠: the numbers multiplying $t$ are the direction ratios.

Similarly, the second line has direction ratios $p', 1, r'$ ⁠.

The lines are perpendicular when

\begin{aligned} &pp' + (1)(1) + rr' = 0 \\ \Rightarrow\ &pp' + rr' = -1 \end{aligned}

Why?

Lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are perpendicular when

a_1a_2 + b_1b_2 + c_1c_2 = 0

Here $a_1, b_1, c_1 = p, 1, r$ and $a_2, b_2, c_2 = p', 1, r'$ ⁠:

\begin{aligned} &(p)(p') + (1)(1) + (r)(r') = 0 \\ &pp' + 1 + rr' = 0 \\ &pp' + rr' = 0 - 1 = -1 \end{aligned}

Watch out: the middle term $(1)(1) = 1$ comes from the $y$ parts. The Assertion leaves it out.

So Assertion (A) is false (it needs $pp' + rr' = -1$ ⁠). Reason (R) is true, since $\vec b_1 \cdot \vec b_2 = 0$ means the directions are perpendicular.

Why?

A counterexample to the Assertion: take $p = p' = 1$ and $r = r' = 0$ ⁠. Then $pp' + rr' = 1$ ⁠, but $pp' + 1 + rr' = 2 \ne 0$ ⁠, so the lines are not perpendicular.

\begin{aligned} pp' + rr' &= (1)(1) + (0)(0) = 1 \\ pp' + 1 + rr' &= 1 + 1 + 0 = 2 \end{aligned}

The Reason is the vector form of the same test: the angle between the lines is the angle between $\vec b_1$ and $\vec b_2$ ⁠. With the Assertion false, the code is (D).

Answer: (D) Assertion (A) is false and Reason (R) is true.

Where this comes from

Chapter 11, Three Dimensional Geometry: Perpendicular lines in space.

Question type: Perpendicular or parallel

How to spot it: click to see

Type 7. Perpendicular or parallel: test the direction ratios

Spot it: "Show that the lines are perpendicular (or parallel)", or "find $k$ so that the lines are at right angles".

Here: Put $y = t$ ⁠: the first line is $(pt + q, t, rt + s)$ ⁠, so the direction ratios are $p, 1, r$ and $p', 1, r'$ ⁠. Perpendicular means $pp' + 1 + rr' = 0$ ⁠, so $pp' + rr' = -1$ ⁠, not 1.

Quick lesson · Worked NCERT solutions · Interactive lesson

Also uses Chapter 10, Vector Algebra.

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Q20: Perpendicular lines x=py+qx = py + qx=py+q⁠, z=ry+sz = ry + sz=ry+s

Q20: Perpendicular lines $x = py + q$ ⁠, $z = ry + s$