Question

The area of the triangle formed by the intersection of a line parallel to x - axis and passing through P(h,k) with the lines y = x and x + y = 2 is $4h^2$. Find the locus of the point P.

Answer 1

$ar\left(△ABC\right)=\frac{1}{2}\left|\begin{array}{ccc}2-k& k& 1\\ k& k& 1\\ 1& 1& 1\end{array}\right|=4h^2$

$\Rightarrow \left|\begin{array}{ccc}2-k& k& 1\\ k& k& 1\\ 1& 1& 1\end{array}\right|=8h^2$

$\Rightarrow (2-k)[k-1-1]-k(k-1)+1(k-k)=4h^2$

$(2-h)(k-2)-k^2+k=4h^2$

$2k-k^2-4+2k-k^2+k=4h^2$

$5k-2k^2-4-4h^2$

$2k^2-5k+4h^2+4=0$

$2y^2-5y+4x^2+4=0$

$\therefore$ Locus P is $4x^2+2y^2-5y+4=0$

Hence, solved.