If the chord of contact of tangents from a point \(A\) to a given circle passes through \(B\), then the…
If the chord of contact of tangents from a point \(A\) to a given circle passes through \(B\), then the circle with \(A B\) as a diameter will ____
Touch the given circle internally
Cut the given circle orthogonally
Touch the given circle externally
Neither intersect nor touch the given circle
Solution
Let an equation of circle
\(x^2+y^2+2 g x+2 f y+c=0 \text { and point } A\left(x_1, y_1\right)\)
Then equation of chord of contact is
\(\begin{aligned}
x x_1+y y_1+g\left(x+x_1\right)+f\left(y+y_1\right)+c & =0 \\
\Rightarrow \quad\left(x_1+g\right) x+\left(y_1+f\right) y+\left(g x_1+f y_1+c\right) & =0
\end{aligned}\)
Let another point \(B\left(x_2, y_2\right)\) through which chord (i) passes, so
\(x_1 x_2+g x_2+y_1 y_2+f y_2+g x_1+f y_1+c=0 \quad \ldots (ii)\)
Now equation of circle having \(A B\) as diameter is
\(\begin{aligned}
& \left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)=0 \\
& \Rightarrow x^2+y^2-\left(x_1+x_2\right) x-\left(y_1+y_2\right) y +x_1 x_2+y_1 y_2=0 \quad \ldots (iii)
\end{aligned}\)
By the circles
\(x^2+y^2+2 g x+2 f y+c=0 \text { and circle (iii) }\)
the \(2 g_1 g_2+2 f_1 f_2\)
\(\begin{aligned}
& =-2 g\left(\frac{x_1+x_2}{2}\right)-2 f\left(\frac{y_1+y_2}{2}\right) \\
& =-g x_1-g x_2-f y_1-f y_2 \\
& =x_1 x_2+y_1 y_2+c \quad \{\text{from Eq. (ii)}\} \\
& =c_1+c_2 \\
\because 2 g_1 g_2 & +2 f_1 f_2=c_1+c_2
\end{aligned}\)
\(\therefore\) Circles \(x^2+y^2+2 g x+2 f y+c=0\) and Eq. (iii) cuts orthogonally.
Hence, option (b) is correct.