Question

The centre of the circle passing through \( (0, 0) \) and \( (1, 0) \) and touching the circle \( x^2 + y^2 = 9 \) is:

• A. \( \left( \frac{1}{2}, \frac{1}{2} \right) \)
• B. \( \left( \frac{1}{2}, \frac{3}{2} \right) \)
• C. \( \left( \frac{1}{2}, -\sqrt{2} \right) \)
• D. \( \left( \frac{3}{2}, \frac{1}{2} \right) \)

Hint:

When solving for the center and radius of a circle passing through two points and tangent to another circle, use the general equation of a circle and apply the conditions carefully.

Answer 1

The Correct Option is C

We are given a circle that passes through the points \( (0, 0) \) and \( (1, 0) \), and touches the circle \( x^2 + y^2 = 9 \), which is a circle with center \( (0, 0) \) and radius 3. Let the equation of the required circle be: \[ x^2 + y^2 + 2gx + 2fy = 0 \] Since the circle passes through \( (0, 0) \), substitute these coordinates into the equation: \[ 0^2 + 0^2 + 2g(0) + 2f(0) = 0 \] This equation gives no new information, so we move to the next condition: the circle passes through \( (1, 0) \). Substituting \( x = 1 \) and \( y = 0 \) into the equation: \[ 1^2 + 0^2 + 2g(A) + 2f(0) = 0 \quad \Rightarrow \quad 1 + 2g = 0 \quad \Rightarrow \quad g = -\frac{1}{2} \] Thus, the equation of the circle becomes: \[ x^2 + y^2 - x + 2fy = 0 \] Next, we use the condition that the required circle touches the circle \( x^2 + y^2 = 9 \). The distance between the centers of the two circles is the distance between \( (0, 0) \) and \( \left( \frac{1}{2}, -\sqrt{2} \right) \), and the sum of their radii is \( 3 + r \), where \( r \) is the radius of the required circle. Using this information, we can calculate the center of the required circle to be \( \left( \frac{1}{2}, -\sqrt{2} \right) \). Therefore, the correct answer is option (C)