Question

Find the equation of locus of a point P which divides segment AQ internally in the ratio $2:5$, where $A(3,-2)$ and Q is any point of the locus $x^2+y^2-4x-12=0$

Answer 1

$\alpha =\frac{2x+5\left(3\right)}{5+2};\beta =\frac{2y+5(-2)}{5+2}$

$\alpha =\frac{2x+15}{7};\beta =\frac{2y-10}{7}$

$xx=\frac{7\alpha -15}{2};y=\frac{7\beta +10}{2}$ ...(1)

$x^2+y^2-4x-12=0$

$\Rightarrow (x-2{)}^2+y^2=16$ ...(2)

Using (1) and (2)

${(\frac{7\alpha -15}{2}-2)}^2+{\left(\frac{7\beta +10}{2}\right)}^2=16$

${\left(\frac{1\alpha -19}{2}\right)}^2+{\left(\frac{7\beta +10}{2}\right)}^2=16$

$\Rightarrow (7\alpha -19{)}^2+(7\beta +10{)}^2=64$

$\therefore$ The equation of P is of the form

$(7\alpha -19{)}^2+(7\beta +10{)}^2=64$