Question

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

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

Answer 1

The correct option is B a parabola

Given that

$\tan A+\tan B=c\dots \dots \left(1\right)$

where c is a constant

From the figure, we have

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

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

Putting it in the equation (1)

$\frac{k}{a+h}+\frac{k}{a-h}=c2ak=c(a^2-h^2)⟹c{x}^2=c{a}^2-2ay$

Hence, locus of the vertex is the parabola.