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 $y=mx\pm \sqrt{16m^2+9}$
$\Rightarrow 4=5m\pm \sqrt{16m^2+9}$
$\Rightarrow {\left(4-5m\right)}^2=16m^2+9\Rightarrow 9m^2-40m+7=0$
$\Rightarrow m_1+m_2=\frac{40}{9},m_1{m}_2=\frac{7}{9}$
$\Rightarrow \left(m_1+m_2\right)-\left(m_1{m}_2\right)=\frac{33}{9}=\frac{11}{3}$