Question

Tangents are drawn from the point $(4,2)$ to the curve $x^2+9y^2=9,$ then the tangent of acute angle between the tangents is

• A. $\frac{3\sqrt{3}}{5\sqrt{17}}$
• B. $\frac{\sqrt{43}}{10}$
• C. $\frac{\sqrt{43}}{5}$
• D. $\sqrt{\frac{3}{17}}$

Answer 1

The correct option is C $\frac{\sqrt{43}}{5}$

Equation of tangent to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ is :
$y=mx\pm \sqrt{9m^2+1}$

If it passes through $(4,2),$ then
$(2-4m{)}^2=9m^2+1\Rightarrow 16m^2-16m+4=9m^2+1\Rightarrow 7m^2-16m+3=0$

If $m_1$ and $m_2$ are the slope of tangents and $\theta$ is the angle between them, then
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{\sqrt{(m_1+m_2{)}^2-4m_1{m}_2}}{1+m_1{m}_2}\right|=\left|\frac{\sqrt{{\left(\frac{16}{7}\right)}^2-4\left(\frac{3}{7}\right)}}{1+\frac{3}{7}}\right|=\frac{\sqrt{172}}{10}=\frac{\sqrt{43}}{5}$