Question

Let the equation of a circle and a parabola be $x^2+y^2-4x-6=0$ and $y^2=9x$ respectively. Then

• A. $(1,-1)$ is a point on the common chord of contact
• B. the equation of the common chord is $y+1=0$
• C. the length of the common chord is $6$
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A the length of the common chord is $6$
C $(1,-1)$ is a point on the common chord of contact

Given parabola and circle are $y^2=9x$ and $x^2+y^2-4x-6=0$ respectively.
Now solving these, $x^2+9x-4x-6=0\Rightarrow x^2+5x-6=0\Rightarrow (x-1)(x+6)=0\Rightarrow x=1,-6$ but $x<0$ is not a solution
Thus $x=1,$ and corresponding $y=3,-3$
Hence point of intersection of the parabola and circle are $(1,-3),(1,3)$
Hence equation of common chord is given by, $x=1$
and length of common chord $=3-(-3)=6$