about to only mathematics

Options

A. hyperbola
B. a parabola
C. an ellipse
D. a straight line

Answer

D

Explanation

The equations of the sides of the triangle are :
`L_(1) -= (x)/(p) - (y)/(1 + p) = -1, L_(2) -= (x)/(q) = (y)/(1+q) = -1 , L_(3) -= y =0`
The coordinates of vertices are `A (-p,0) , B(-q,0) and C(pq , (1 + p) (1+q))`.

The equation of the altitude through C is x = pq and the equation of the altitude through B is (1+p) y + px + pq = 0.
Solving these equations , we get x = pq , y = -pq
Let (h,k) be the coordinates of the orthocentre . Then ,
h = pq and k = -pq `implies` k = - h
Hence , the locus of (h,k) is y = -x , which is a straight line .