Question

Let a variable point $A$ lie on a line $L=0$. Tangents are drawn to the circle $x^2+y^2=9$ through the variable point $A$ meet the circle at $B$ and $D$. A point $C$ lies on the other side of the circle and also on the line $L=0$ such that $ABCD$ is a parallelogram. On the basis of the above information, answer the following question.

If $A=(3,6),$ then co-ordinate of $C$ is

• A. $(-3,6)$
• B. $(3,-6)$
• C. $(-6,-3)$
• D. $(-3,-6).$

Answer 1

Answer: (D)

$(-3,-6).$

Length of tangents from a point (outside the circle) are equal in length

$\therefore AB=AD$ then the parallelogram $ABCD$ is a rhombus.

$\Rightarrow$ Mid point of $BD=$ mid point of $AC$

$\Rightarrow$ $C$ is mirror image of point $A$ w.r.t to $BD$, Also $BD\perp AC$

$\because A=(3,6)$

$\therefore$ Equation of chord of contact through tangents is

$3x+6y-9=0$

Let $C=\alpha ,\beta$

$\Rightarrow C$ is the mirror image of point $A(3,6)$ w. r to the line $3x+6y-9=0$

$\Rightarrow \frac{\alpha -3}{3}=\frac{\beta -6}{6}=\frac{-2\left(3\right(3)+6(6\left)\right)}{(3^2+6^2)}$

$\Rightarrow \frac{\alpha -3}{3}=\frac{\beta -6}{6}=-2$

$\Rightarrow \alpha =-3,\beta =-6$

$\therefore C=(-3,-6)$

Hence choice (D) is correct.