Question

If the length of tangent drawn from the point $(5,3)$ to the circle $x^2+y^2+2x+ky+17=0$ be $7$, then $k=$

• A. $4$
• B. $-4$
• C. $-6$
• D. $\frac{13}{2}$

Answer 1

The correct option is B $-4$

Solution:- (B) $-4$

As we know that length of a tangent $\left(l\right)$ is given by-

$l=\sqrt{S_1}$

$\therefore$ Length of tangent from the point $(5,3)$ to the circle $x^2+y^2+2x+ky+17=0$ will be-

$l=\sqrt{5^2+3^2+2\times 5+k\times 3+17}$

$\Rightarrow l=\sqrt{25+9+10+3k+17}$

$\Rightarrow l=\sqrt{61+3k}$

Given that length of tangent is $7$.

$\therefore \sqrt{61+3k}=7$

Squaring both sides, we have

$\Rightarrow 61+3k=49$

$\Rightarrow -3k=12=\Rightarrow k=-4$

Hence the correct answer is $-4$.