JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $P$ be the plane passing through the points $(5, 3, 0), (13, 3, -2)$ and $(1, 6, 2)$ . For $\alpha \in \mathbb{N}$ , if the distances of the points $A(3, 4, \alpha)$ and $B(2, \alpha, a)$ from the plane $P$ are $2$ and $3$ , respectively, then the positive value of $a$ is:

Choose the correct answer:

Explanation

Solution

1. Plane ki Equation:

Points $(5,3,0), (13,3,-2), (1,6,2)$ ka upyog karke plane ki equation:

\begin{vmatrix} x-5 & y-3 & z-0 \\ 13-5 & 3-3 & -2-0 \\ 1-5 & 6-3 & 2-0 \end{vmatrix} = 0 \implies \begin{vmatrix} x-5 & y-3 & z \\ 8 & 0 & -2 \\ -4 & 3 & 2 \end{vmatrix} = 0

Equation: $6(x-5) - 8(y-3) + 24z = 0 \implies 3x - 4y + 12z - 3 = 0$

2. Point A(3, 4, $\alpha$ ) se distance:

d_1 = \frac{|3(3) - 4(4) + 12\alpha - 3|}{\sqrt{3^2 + (-4)^2 + 12^2}} = 2 \implies \frac{|12\alpha - 10|}{13} = 2

12\alpha - 10 = 26 \implies \alpha = 3 \ (\text{as } \alpha \in \mathbb{N})

3. Point B(2, $\alpha$ , a) yaani B(2, 3, a) se distance:

d_2 = \frac{|3(2) - 4(3) + 12a - 3|}{13} = 3 \implies |12a - 9| = 39

12a - 9 = 39 \implies 12a = 48 \implies a = 4

Sahi Answer: $a = 4$