In triangle ABC, base BC is fixed. Then prove that the locus of vertex A such that tan B+tan C= Constant is parabola.

Explanation

Let base BC lie on x-axis with B at (0,0) and C at (a,0). Here, a is constant.
Now, according to the question,
`tanalpha+tanbeta=c" (constant)"`
`rArr" "(y)/(x)+(y)/(a-x)=c`
`rArr" "(ay)/(x(a-x))=c`
`rArr" "ay=c(ax-x^(2))`, which is equation of locus of vertex A.
Clearly, this is equation of parabola.