Let O be the mid point of base BC of triangle ABC and the vertex A be (h,k)
In △ABDtanx=kh+a....(i)
In △ACDtanϕ=kh−a.....(ii)
ϕ+y=π⇒y=π−ϕ
Given tanxtany=λ
tanxtan(π−ϕ)=λtanx(−tanϕ)=λ−tanxtanϕ=λ
substituting (i) and (ii)
−kh+a×kh−a=λk2h2−a2=−λk2=−λh2+a2λk2+λh2=a2λ
Replacing h by x and y by k
y2+λx2=a2λ
is the required locus of vertex A