Question

The locus of the point of intersection of tangents to an ellipse at two points, sum of whose eccentric angles is constant is a/an:

• A. parabola
• B. circle
• C. ellipse
• D. straight line

Answer 1

The correct option is D straight line

The equations of tangents at two points having eccentric angles ${\theta }_1$ and ${\theta }_2$ are
$\frac{x}{a}cos{\theta }_1+\frac{y}{b}sin{\theta }_1=1$ .... (i)
and $\frac{x}{a}cos{\theta }_2+\frac{y}{b}sin{\theta }_2=1$ ..... (ii)
The point of intersection of (i) and (ii) is
$[\frac{acos\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}{cos\left(\frac{{\theta }_1-{\theta }_2}{2}\right)},\frac{bsin\left(\frac{{\theta }_1+{\theta }_2}{2}\right)}{cos\left(\frac{{\theta }_1-{\theta }_2}{2}\right)}]$
It is given that ${\theta }_1+{\theta }_2=c=$ constant.

$\frac{{\theta }_1+{\theta }_2}{2}=\frac{c}{2}=k,$ $k$ is also a constant.

Therefore, if $(x_1,y_1)$ is the point of intersection of (i) and (ii), then
$x_1=\frac{acosk}{cos\left(\frac{{\theta }_1-{\theta }_2}{2}\right)}$
and $y_1=\frac{bsink}{cos\left(\frac{{\theta }_1-{\theta }_2}{2}\right)}$
$\Rightarrow \frac{x_1}{y_1}=\frac{a}{b}cotk\Rightarrow y_1=\left(\frac{b}{a}tank\right)x_1$
$\Rightarrow (x_1,y_1)$ lies on the straight line $y=\left(\frac{b}{a}tank\right)x$.