JEE Main 2025 Apr 3 Shift 2, Mathematics Q5: Introduction to Straight Lines

JEE Main2025Apr 3, Shift 2Mathematics

Consider the lines $x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5$ , $λ$ being a parameter, all passing through a point $P$ . One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$ , then the value of $d^{2}$ is

A
$20$
B
$30$
C
$10$
D
$15$

Solution

$x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5$

$\Rightarrow (x + 2 y - 5) + λ (3 x + 7 y - 17) = 0$

This represents a family of lines.

Point of intersection:

$P (1, 2)$

Let $Q (3, 6)$ .

$d = P Q = (3 - 1)^{2} + (6 - 2)^{2} = 2^{2} + 4^{2} = 20$

$∴ d^{2} = 20$

$Option (1)$

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JEE Main 2025 Apr 3 Shift 2, Mathematics Q5: Introduction to Straight Lines

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