Question

The angle between the tangents drawn at the points $(5,12)$and $(12,-5)$ to the circles $x^2+y^2=169$ is

• A. ${45}^{°}$
• B. ${60}^{°}$
• C. ${30}^{°}$
• D. ${90}^{°}$

Answer 1

The correct option is D

${90}^{°}$

Explanation for the correct option:

Given the equation of the circle : $x^2+y^2=169$

Differentiating with respect to $x$ we get,

$$\begin{gathered} 2x+2y\frac{dy}{dx}=0 \\ \frac{dy}{dx}=-\frac{x}{y} \end{gathered}$$

For $m_1$

${\left[\frac{dy}{dx}\right]}_{(5,12)}=-\frac{5}{12}$

For $m_2$

${\left[\frac{dy}{dx}\right]}_{(12,-5)}=\frac{12}{5}$

We know that if $m_1{m}_2=-1$ then the lines will be perpendicular, so

$\therefore \left(-\frac{5}{12}\right)\left(\frac{12}{5}\right)=-1$

This proves that the angle between the tangents is ${90}^{°}$

Hence, the correct option is (D).