Question

The centre of the circle whose radius is 3 units and touching internally the circle $x^2 + y^2 - 4x - 6y - 12 = 0$ at the point $(-1, -1)$ is

• A. $(4/5, 7/5)$
• B. $(4/5, -7/5)$
• C. $(-4/5, -7/5)$
• D. $(-4/5, 7/5)$

Hint:

For circles touching internally, the point of tangency lies on the line joining the centers and divides the segment formed by the centers externally in the ratio of their radii. The distance between centers is $|r_1 - r_2|$.

Answer 1

The Correct Option is A

Step 1: Find the center and radius of the given circle. The equation of the given circle is $x^2 + y^2 - 4x - 6y - 12 = 0$. Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = -4 \Rightarrow g = -2$ $2f = -6 \Rightarrow f = -3$ $c = -12$ The center of the given circle, $C_1$, is $(-g, -f) = (2, 3)$. The radius of the given circle, $r_1$, is $\sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-3)^2 - (-12)}$: \[ r_1 = \sqrt{4 + 9 + 12} = \sqrt{25} = 5 \]
Step 2: Define properties of the new circle. Let the new circle be $C_2$ with center $(h, k)$ and radius $r_2 = 3$ (given). The two circles touch internally at the point $P(-1, -1)$.
Step 3: Apply the condition for internal tangency. When two circles touch internally, the point of tangency $P$ lies on the line segment joining their centers, and $P$ divides the line segment $C_1C_2$ externally in the ratio of their radii, $r_1 : r_2$. Here, $C_1 = (2, 3)$, $C_2 = (h, k)$, $P = (-1, -1)$, $r_1 = 5$, $r_2 = 3$. So, $P$ divides $C_1C_2$ externally in the ratio $5:3$.
Step 4: Use the external section formula to find $(h, k)$. The coordinates of a point $P(x, y)$ that divides the line segment joining $C_1(x_1, y_1)$ and $C_2(x_2, y_2)$ externally in the ratio $m:n$ are given by: \[ (x, y) = \left( \frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n} \right) \] Using $P(-1, -1)$, $C_1(2, 3)$, $C_2(h, k)$, $m=5$, $n=3$: For the x-coordinate: \[ -1 = \frac{5(h) - 3(2)}{5 - 3} \] \[ -1 = \frac{5h - 6}{2} \] \[ -2 = 5h - 6 \] \[ 5h = 4 \Rightarrow h = \frac{4}{5} \] For the y-coordinate: \[ -1 = \frac{5(k) - 3(3)}{5 - 3} \] \[ -1 = \frac{5k - 9}{2} \] \[ -2 = 5k - 9 \] \[ 5k = 7 \Rightarrow k = \frac{7}{5} \]
Step 5: State the center of the new circle. The center of the circle whose radius is 3 units and touching internally the given circle at $P(-1, -1)$ is $(h, k) = (4/5, 7/5)$.