Question

A line is drawn through point P (4, 7) which cuts the circle ${{x}^{2}}+{{y}^{2}}=9$ at the points A and B. Then, PA.PB is equal to:
(a) 74
(b) 56
(c) 53
(d) 65

Answer 1

Hint: There is a property in the circle that if a line drawn through certain point P $\left( {{x}_{1}},{{y}_{1}} \right)$ then and that line cuts the circle in two points A and B then multiplication of PA and PB is equal to the power of the point P $\left( {{x}_{1}},{{y}_{1}} \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$. Now, we know that power of the point say P ($\left( {{x}_{1}},{{y}_{1}} \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$ is equal to ${{S}_{1}}$ which is equal to ${{x}_{1}}^{2}+{{y}_{1}}^{2}-{{a}^{2}}$ so similarly, the multiplication of PA and PB is equal to the power of circle ${{x}^{2}}+{{y}^{2}}=9$ with respect to the point P (4, 7).

Complete step-by-step solution:
We have given that a line is drawn through point P (4, 7) which cuts the circle ${{x}^{2}}+{{y}^{2}}=9$ into two points A and B. And we have to find PA.PB.
In the below figure, we have drawn a circle ${{x}^{2}}+{{y}^{2}}=9$ with point P (4, 7) is cutting the circle into two points A and B.

To find the multiplication of PA and PB we are going to use the following property of the circles which say that, if a line drawn through certain point P $\left( {{x}_{1}},{{y}_{1}} \right)$ then and that line cuts the circle in two points A and B then multiplication of PA and PB is equal to the power of the point P $\left( {{x}_{1}},{{y}_{1}} \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$. Now, we know that power of the point say P ($\left( {{x}_{1}},{{y}_{1}} \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$ is equal to ${{S}_{1}}$ which is equal to ${{x}_{1}}^{2}+{{y}_{1}}^{2}-{{a}^{2}}$.
Now, to find the multiplication of PA and PB we are going to calculate the power of a circle ${{x}^{2}}+{{y}^{2}}=9$ with respect to P (4, 7) we get,
$\begin{align}
  & {{S}_{1}}={{4}^{2}}+{{7}^{2}}-9 \\
 & \Rightarrow {{S}_{1}}=16+49-9 \\
 & \Rightarrow {{S}_{1}}=16+40 \\
 & \Rightarrow {{S}_{1}}=56 \\
\end{align}$
Hence, the multiplication of PA and PB is equal to 56. Hence, the correct option is (b).

Note: This problem brushes our concept of the circles that if a line is drawn through any point say P cuts the circle in two different points PA and PB then the product of PA and PB are the power of the circle with respect to point P. It would be better if you can memorize this concept it will save your time in the examination.