Question

If $m_{1}$ and $m_{2}$ are the slopes of the tangents to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ which passes through $\left(5,4\right),$ then the value of $\left(m_{1} + m_{2}\right)-\left(m_{1} m_{2}\right)$ is equal to

• A. $\frac{47}{9}$
• B. $-\frac{40}{9}$
• C. $\frac{22}{3}$
• D. $\frac{11}{3}$

Answer 1

Answer: (D)

Let the equation of tangent passing through $\left(5,4\right)$ is $y=mx\pm\sqrt{16 m^{2} + 9}$
$\Rightarrow 4=5m\pm\sqrt{16 m^{2} + 9}$
$\Rightarrow \left(4 - 5 m\right)^{2}=16m^{2}+9\Rightarrow 9m^{2}-40m+7=0$
$\Rightarrow m_{1}+m_{2}=\frac{40}{9},m_{1}m_{2}=\frac{7}{9}$
$\Rightarrow \left(m_{1} + m_{2}\right)-\left(m_{1} m_{2}\right)=\frac{33}{9}=\frac{11}{3}$