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
(©) an ellipse
(d) a hyperbola​ i need full explanation

Answer 1

Answer:

first one is correct because incircle of a base angle is constant

Answer 2

The locus of the vertex is a parabola

= Tan A + Tan B = C  --- eq 1

Tan A = k / a - h and

Tan B = k / a + h

Substituting in equation 1 -

= {k/(h+a)} + {k/(h-a)} = C

= 2ak = c ( a² - h²)

= cx² = ca² - 2ay

Thus, the locus of the vertex is a parabola.