Question

The coordinates of the points A and B are (-2, 3) and (4, -5) respectively. If a point P moves so that PA2 - PB2 = 20 , then the equation to the locus of the point P is​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{A(-2,3),\;B(4,-5)\;and\;PA^2-PB^2=20}$

$\underline{\textbf{To find:}}$

$\textsf{Locus of the point P}$

$\underline{\textbf{Solution:}}$

$\textsf{Let the coordinates of the moving point P be (h,k)}$

$\textsf{As per given data,}$

$\mathsf{PA^2-PB^2=20}$

$\mathsf{(\sqrt{(h+2)^2+(k-3)^2})^2-(\sqrt{(h-4)^2+(k+5)^2})^2=20}$

$\mathsf{[(h+2)^2+(k-3)^2]-[(h-4)^2+(k+5)^2]=20}$

$\mathsf{[h^2+4+4h+k^2+9-6k]-[h^2+16-8h+k^2+25+10k]=20}$

$\mathsf{h^2+4h+k^2+13-6k-h^2+8h-k^2-41-10k=20}$

$\mathsf{4h+13-6k+8h-41-10k=20}$

$\mathsf{12h-16k-48=0}$

$\mathsf{3h-4k-12=0}$

$\therefore\textsf{The locus of P is}$

$\textbf{3x-4y-12=0}$