ISI2017-DCG-19

Question

The angle between the tangents drawn from the point $(-1, 7)$ to the circle $x^2+y^2=25$  is

• A. $\tan^{-1} (\frac{1}{2})$
• B. $\tan^{-1} (\frac{2}{3})$
• C. $\frac{\pi}{2}$
• D. $\frac{\pi}{3}$

Comments

I am getting $m = \frac{-4}{3}, \frac{3}{4}$

Answer 1

The equation of tangent to a circle(in slope form) is given by :  $y=mx+a\sqrt{1+m^{2}}$

Tangent(s) pass through the point   $\left ( -1,7 \right )$  

So, putting the values in tangent equation, we get  $7=m\left ( -1 \right )+5\sqrt{1+m^{2}}$

$\Rightarrow$    $m+7=5\sqrt{1+m^{2}}$

$\Rightarrow$    $24m^{2}-14m-24=0$

Let the roots of this equation be $m_{1},m_{2}$.

So, from the quadratic equation, we see that  $m_{1}\times m_{2}=-1$    (product of roots is  $\frac{c}{a}$)

$\Rightarrow$    Tangents are perpendicular to each other

Option C is correct.