Sample 2022-23 Class 12 Maths Q20 Solution: Acute angle between a line and the x-axis

The question

In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.

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

(b) Both A and R are true but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

Assertion (A): The acute angle between the line $\bar r = \hat i + \hat j + 2\hat k + \lambda(\hat i - \hat j)$ and the x-axis is $\frac{\pi}{4}$

Reason(R): The acute angle $\theta$ between the lines $\bar r = x_1\hat i + y_1\hat j + z_1\hat k + \lambda(a_1\hat i + b_1\hat j + c_1\hat k)$ and $\bar r = x_2\hat i + y_2\hat j + z_2\hat k + \mu(a_2\hat i + b_2\hat j + c_2\hat k)$ is given by $\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$

Exam answer

The x-axis is $\vec r = t\hat i$ ⁠, with direction ratios $1, 0, 0$ ⁠. The line has direction ratios $1, -1, 0$ ⁠.

Why?

In $\vec r = \vec a + \lambda\vec b$ ⁠, the vector $\vec b$ multiplying the parameter gives the direction. For the given line $\vec b = \hat i - \hat j$ ⁠, that is $1\hat i - 1\hat j + 0\hat k$ ⁠.

The x-axis passes through the origin in the direction $\hat i$ ⁠, so it is $\vec r = t\hat i$ ⁠.

\begin{aligned} &a_1a_2 + b_1b_2 + c_1c_2 \\ &= (1)(1) + (-1)(0) + (0)(0) = 1 \end{aligned}

\begin{aligned} \cos\theta &= \frac{|1|}{\sqrt{1 + 1 + 0}\,\sqrt{1 + 0 + 0}} \\ &= \frac{1}{\sqrt 2} \end{aligned}

Why?

This is the formula of the Reason, with $a_1, b_1, c_1 = 1, -1, 0$ and $a_2, b_2, c_2 = 1, 0, 0$ ⁠. The denominators are the lengths $\sqrt{1^2 + (-1)^2 + 0^2} = \sqrt 2$ and $\sqrt{1^2} = 1$ ⁠.

\begin{aligned} &\text{Top: } (1)(1) + (-1)(0) + (0)(0) \\ &= 1 + 0 + 0 = 1 \\[4pt] &1^2 + (-1)^2 + 0^2 \\ &= 1 + 1 + 0 = 2 \\[4pt] &1^2 + 0^2 + 0^2 = 1 \\[4pt] &\cos\theta = \frac{1}{\sqrt 2 \times \sqrt 1} = \frac{1}{\sqrt 2} \end{aligned}

The modulus in the numerator makes $\cos\theta \ge 0$ ⁠, so $\theta$ is the acute angle.

$\therefore\ \theta = \frac{\pi}{4}$ ⁠, so Assertion (A) is true.

Why?

$\cos\frac{\pi}{4} = \frac{1}{\sqrt 2}$ ⁠, and $\frac{\pi}{4}$ is the only angle from 0 to $\frac{\pi}{2}$ with this cosine.

Reason (R) is the formula for the acute angle between two lines, so it is true.

It is the formula used above to find the angle, so it explains (A).

Why?

For code (a), both statements must be true and the Reason must be the actual reason the Assertion holds. Here the Assertion's angle comes straight from the Reason's formula, so (a), not (b).

Answer: (a) Both A and R are true and R is the correct explanation of A.

Where this comes from

Chapter 11, Three Dimensional Geometry: Angle between two lines.

Question type: The angle between two lines

How to spot it: click to see

Type 6. The angle between two lines: only the directions matter

Spot it: "Find the angle between the lines". The points on the lines play no part.

Here: The x-axis has direction ratios $1, 0, 0$ and the line has $1, -1, 0$ ⁠; its point $\hat i + \hat j + 2\hat k$ plays no part. So $\cos\theta = \frac{|1|}{\sqrt2 \times 1}$ and $\theta = \frac{\pi}{4}$ ⁠.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Q20: Acute angle between a line and the x-axis