Geo5.2.27

Find the pole of the line $3x-5y-9=0\,$ w.r.t the ellipse $4x^2+8y^2-16x+15=0\,$

Let the pole of the given line w.r.t the given ellipse be $P(x_1,y_1)\,$

Equation of polar of P is

$4xx_1+8yy_1-8(x+x_1)+15=0\,$

$x(4x_1-8)+8yy_1+15-8x_1=0\,$

Comparing this equation with $3x-5y-9=0\,$

$\frac{4x_1-8}{3}=\frac{8y_1}{-5}=\frac{15-8x_1}{-9}\,$

$\frac{4x_1-8}{3}=\frac{15-8x_1}{-9},\frac{8y_1}{-5}=\frac{15-8x_1}{-9}\,$

$-36x_1+72=45-24x_1,-72y_1=-75+40x_1\,$

$x_1=\frac{27}{12}=\frac{9}{4},-72y_1=-75+90,y_1=\frac{5}{-24}\,$

Hence the pole of the given line is $\left(\frac{9}{4},\frac{-5}{24}\right)\,$