Question

Consider the circles $C_1: x^2 + y^2 = 16$ and $C_2: x^2 + y^2 - 12x + 32 = 0$. Which of the following statement is/are correct?

• A. Number of common tangent to these circles is 3.
• B. The point P with coordinates (4,1) lies outside the circle $C_1$ and inside the circle $C_2$.
• C. Their direct common tangent intersect at (12,0).
• D. Slope of their radical axis is not defined.

Answer 1

Answer: (A), (C), (D)

(A), (C), and (D)

Answer 2

Circle $C_1$: $x^2 + y^2 = 16$ has center $O_1 = (0,0)$ and radius $r_1 = 4$.

Circle $C_2$: $x^2 + y^2 - 12x + 32 = 0$. Completing the square: $(x-6)^2 + y^2 = 4$. This circle has center $O_2 = (6,0)$ and radius $r_2 = 2$.

The distance between the centers $O_1$ and $O_2$ is $d = \sqrt{(6-0)^2 + (0-0)^2} = 6$.

The sum of the radii is $r_1 + r_2 = 4 + 2 = 6$. Since $d = r_1 + r_2$, the circles touch externally.

Let's analyze each statement:

(A) Number of common tangents: When two circles touch externally, they have exactly 3 common tangents (2 direct and 1 transverse). So, statement (A) is correct.

(B) Position of point P(4,1): For $C_1$: $4^2 + 1^2 - 16 = 16 + 1 - 16 = 1$. Since $1 > 0$, P is outside $C_1$. For $C_2$: $(4-6)^2 + 1^2 - 4 = (-2)^2 + 1 - 4 = 4 + 1 - 4 = 1$. Since $1 > 0$, P is outside $C_2$. The statement claims P is outside $C_1$ (true) and inside $C_2$ (false). So, statement (B) is incorrect.

(C) Intersection of direct common tangents: The direct common tangents intersect at the exsimilicenter, which divides the line segment joining the centers externally in the ratio of their radii ($r_1:r_2 = 4:2 = 2:1$). Using the external division formula for $O_1(0,0)$ and $O_2(6,0)$: $T = \left( \frac{2 \times 6 - 1 \times 0}{2-1}, \frac{2 \times 0 - 1 \times 0}{2-1} \right) = \left( \frac{12}{1}, \frac{0}{1} \right) = (12,0)$. So, statement (C) is correct.

(D) Slope of the radical axis: The radical axis is found by $C_1 - C_2 = 0$. $(x^2 + y^2 - 16) - (x^2 + y^2 - 12x + 32) = 0$ $12x - 48 = 0$ $x = 4$. This is a vertical line, whose slope is undefined. So, statement (D) is correct.

The correct statements are (A), (C), and (D).