Circles
Circle
Allen Star Batch
Grade 11
Question:
If the two circles $C_1: x^2 + y^2 = 16$ and circle $C_2$ of radius $5$ units intersect in such a manner that the common chord of maximum length has a slope equal to $3/4$, then the coordinates of the centre of $C_2$ are:
$\left(\pm\frac{9}{5}; \pm\frac{12}{5}\right)$
$\left(\pm\frac{9}{5}; \pm\frac{12}{5}\right)$
$\left(\pm\frac{12}{5}; \pm\frac{9}{5}\right)$
$\left(\pm\frac{12}{5}; \pm\frac{9}{5}\right)$
Step-by-Step Solution
Key Concept: The angle between two radii determines the slope of the line joining centers, giving discrete solutions.
The slope of $C_1C_2$ is $-\frac{4}{3} = \tan\theta$. Using the parametric form with $\tan\theta_1 = \frac{3}{5}$ and $\tan\theta_2 = \frac{4}{5}$, we find $C_1 = \left(\frac{9}{5}, \frac{12}{5}\right)$ or $\left(0, -3\right)$ and $C_2 = \left(\frac{9}{5}, -\frac{12}{5}\right)$ or $\left(0, 3\right)$. These represent the two possible positions of the centers satisfying the slope condition.
Correct Answer: 2