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 (Bi a parabola (C) an ellipse (D) a hyperbola

Answer 1

AB=2a
`tanalpha+tanbeta=C`
`k/(a+n)+k/(a-h)=C`
`(k(a-h)+k(a+h))/(a^2-h^2)=C`
`ka-kh+ka+kh=C(a^2-h^2)`
`2ka=Ca^2-Ch^2`
`y*(2a)/C=a^2-x^2`
`x^2=(-2ay)/C+a^2`
`x^2=-a((2y)/C-a)`
`x^2=4ay`(Parabola).