Question

If on a given base, a triangle be inscribed such that the sum of the tangents of the base angles is a constant, then the locus of the vertex is _____________.

• A. circle
• B. parabola
• C. ellipse
• D. hyperbola

Answer 1

The correct option is B parabola

$\tan A=\frac{k}{a-h}\tan B=\frac{k}{a+h}$

$\tan A+\tan B=c$ where $c$ is a constant.

$\frac{k}{a-h}+\frac{k}{a+h}=c$

$\Rightarrow k\left[\frac{a+h+a-h}{a^2-h^2}\right]=c$

$\Rightarrow a^2-h^2=\frac{2a}{c}.k$

$\Rightarrow$Locus of $(h,k)$ will be

$a^2-x^2=\frac{2a}{c}.y$

$x^2=a^2-\frac{2a}{c}.y$ is a required parabola.