Q46 / 110 · VITEEE 2024 · Mathematics

If the tangent at the point $P$ on the circle $x^2+y^2+2 x+2 y=7$ meets the straight line $3 x-4 y=15$ at the point $Q$ on the $X$-axis, then length of $P Q$ is

  • A
    $3 \sqrt{7}$
  • B
    $4 \sqrt{7}$
  • Correct answer
    C
    $2 \sqrt{7}$
  • D
    $\sqrt{7}$

Solution

$\because Q$ is on the $X$-axis. Let, the coordinates of $Q=(h, 0)$ Then, $(h, 0)$ lies on the line. Therefore, $3 h-4 \times 0=15 \Rightarrow h=5$ That means, coordinates of $Q=(5,0)$ We know, the length of tangent $=\sqrt{S_1}$ and $S_1=(5)^2+(0)^2+2 \times 5+2 \times 0-7$ $=35-7=28$ Hence, the length of $P Q=\sqrt{28}=2 \sqrt{7}$

Hence, the correct answer is option: 'C'.

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