Board 2023 Class 12 Maths Q20 Solution: Line through (1, 2, 3) and (3, −1, 3)

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 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) : Equation of a line passing through the points $(1, 2, 3)$ and $(3, -1, 3)$ is $\frac{x - 3}{2} = \frac{y + 1}{3} = \frac{z - 3}{0}$ ⁠.

Reason (R) : Equation of a line passing through points $(x_1, y_1, z_1)$ ⁠, $(x_2, y_2, z_2)$ is given by $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$ ⁠.

Exam answer

Direction ratios of the line: $3 - 1, -1 - 2, 3 - 3$ ⁠, that is $2, -3, 0$ ⁠.

Why?

For a line through two points, the direction ratios are the differences of the coordinates:

x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1

Second point minus first point, $(3, -1, 3) - (1, 2, 3)$ ⁠:

\begin{aligned} 3 - 1 &= 2 \\ -1 - 2 &= -3 \\ 3 - 3 &= 0 \end{aligned}

Watch the middle one: $-1 - 2 = -3$ ⁠, not 3.

The line through $(3, -1, 3)$ is

\frac{x - 3}{2} = \frac{y + 1}{-3} = \frac{z - 3}{0}

Why?

Any point of the line can be used in the numerators. With $(3, -1, 3)$ ⁠, $y - (-1)$ becomes $y + 1$ ⁠.

A 0 in a denominator is allowed here: it means $z$ does not change, so $z = 3$ all along the line.

In the Assertion, check the point $(1, 2, 3)$ ⁠:

\frac{1 - 3}{2} = -1,\quad \frac{2 + 1}{3} = 1

Since $-1 \ne 1$ ⁠, $(1, 2, 3)$ is not on the Assertion's line. So Assertion (A) is false (the second denominator is $-3$ ⁠, not 3).

Reason (R) is true: it is the two-point form of a line.

Why?

Put $x = 1$ ⁠, $y = 2$ into the Assertion's first two fractions:

\begin{aligned} \frac{1 - 3}{2} &= \frac{-2}{2} = -1 \\ \frac{2 + 1}{3} &= \frac{3}{3} = 1 \end{aligned}

A line through both points must contain $(1, 2, 3)$ ⁠, so its fractions must be equal there. The Assertion's line has direction ratios $2, 3, 0$ ⁠, which are not proportional to $2, -3, 0$ ⁠.

The Reason states the standard two-point form correctly. The Assertion is false and the Reason is true, and only code (d) says this.

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

Where this comes from

Chapter 11, Three Dimensional Geometry: Equation of a line through two points.

Question type: The line through two points

How to spot it: click to see

Type 2. The line through two points: subtract, then divide

Spot it: two points are given, and you need the direction of the line through them.

Here: Subtract $(1, 2, 3)$ from $(3, -1, 3)$ to get the direction ratios $2, -3, 0$ ⁠. The middle bottom must be $-3$ ⁠, so the Assertion's 3 is wrong while the Reason's two-point form is right.

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Q20: Line through $(1, 2, 3)$ and $(3, -1, 3)$