Question

The point of which the line $9x+y-28=0$ is the chord of contact of the circle $2x^2+2y^2-3x+5y-7=0$ is?

• A. $(3,-1)$
• B. $(3,1)$
• C. $(-3,1)$
• D. None of these

Answer 1

Answer: (A)

$(3,-1)$

For any general circle $x^2+y^2+2gx+2fy+c=0$, the chord of contact of tagents from $(x_1,y_1)$ is given as $x_{1}x+y_{1}y+g(x_1+x)+f(y_1+y)+c=0$

Hence, comparing $x^2+y^2+2gx+2fy+c=0$ with $2x^2+2y^2-3x+5y-7=0⟹x^2+y^2-\frac{3}{2}x+\frac{5}{2}y-\frac{7}{2}=0$ gives $g=-\frac{3}{4},f=\frac{5}{4}$ and $c=-\frac{7}{2}$

$\therefore x_{1}x+y_{1}y+g(x_1+x)+f(y_1+y)+c=x_{1}x+y_{1}y-\frac{3}{4}(x_1+x)+\frac{5}{4}(y_1+y)-\frac{7}{2}=0$

$\therefore 4x_{1}x+4y_{1}y-3(x_1+x)+5(y_1+y)-14=0$

$\therefore 4x_{1}x+4y_{1}y-3(x_1+x)+5(y_1+y)-14=9x+y-28$

on comparing coefficients of $x$, we have

$4x_1-3=9⟹x_1=3$

on comparing coefficients of $y$, we have

$4y_1+5=1⟹y_1=-1$

Hence, required point is $(3,-1)$.