Question

Let $ S $ be a circle concentric with the circle $ 3x^2 + 3y^2 + x + y - 1 = 0 $. If the length of the tangent drawn from a point $ (2, -2) $ to the given circle is the radius of the circle $ S $, then the power of the point $ (2, 1) $ with respect to the circle $ S $ is

• A. \( \frac{-137}{18} \)
• B. \( \frac{1}{18} \)
• C. \( \frac{-29}{18} \)
• D. \( \frac{23}{18} \)

Hint:

The power of a point with respect to a circle can be calculated using the general formula involving the coordinates of the point and the equation of the circle. Don't forget to complete the square when necessary.

Answer 1

The Correct Option is C

The given equation of the circle is: \[ 3x^2 + 3y^2 + x + y - 1 = 0 \] This can be simplified as: \[ x^2 + y^2 + \frac{x}{3} + \frac{y}{3} - \frac{1}{3} = 0 \] Step 1: The radius of this circle can be found by completing the square for both \( x \) and \( y \). Complete the square for \( x \) and \( y \) terms: \[ \left(x + \frac{1}{6}\right)^2 + \left(y + \frac{1}{6}\right)^2 = \frac{1}{18} \] Thus, the radius of the given circle is \( r = \frac{1}{\sqrt{18}} \).
Step 2: The power of a point \( (x_1, y_1) \) with respect to a circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) is given by: \[ \text{Power} = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c \] For the point \( (2, 1) \) and the circle \( 3x^2 + 3y^2 + x + y - 1 = 0 \), substituting the values and calculating the power gives: \[ \text{Power} = \frac{-29}{18} \] Thus, the power of the point \( (2, 1) \) with respect to the circle \( S \) is \( \frac{-29}{18} \).