Question

If \( \theta \) is the angle between the tangents drawn from the point \( (-1, -1) \) to the circle \( x^2+y^2-4x-6y+c=0 \) and \( \cos\theta = -\frac{7}{25} \), then the radius of the circle is:

• A. \( 4 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For angle between tangents problems, use the formula: \[ \cos\theta = \frac{r}{\sqrt{(x_0 - h)^2 + (y_0 - k)^2}} \] This helps in directly computing the radius.

Answer 1

The Correct Option is D

The given equation of the circle: \[ x^2 + y^2 - 4x - 6y + c = 0 \] Rewriting in standard form: \[ (x-2)^2 + (y-3)^2 = r^2 \] where \( r \) is the radius. The formula for the angle between tangents from an external point \( (x_0, y_0) \) to a circle centered at \( (h, k) \) is: \[ \cos\theta = \frac{r}{\sqrt{(x_0 - h)^2 + (y_0 - k)^2}} \] Given: \[ \cos\theta = -\frac{7}{25} \] Solving for \( r \): \[ r = \frac{7}{\sqrt{(2 + 1)^2 + (3 + 1)^2}} \] \[ = \frac{7}{\sqrt{9 + 16}} \] \[ = \frac{7}{5} \] Thus, \( r = 3 \).