Question

If the inverse point of the point $ (3, 2) $ with respect to the circle $ x^2 + y^2 - 2x + 4y - 4 = 0 $ is $ (l, m) $, then $ 2l + 19m = $:

• A. 3
• B. 1
• C. 0
• D. -1

Hint:

To find the inverse point with respect to a circle, use the formula involving the radius and the distance from the center to the point.

Answer 1

The Correct Option is C

The equation of the circle is \( x^2 + y^2 - 2x + 4y - 4 = 0 \). Completing the square for \( x \) and \( y \): \[ (x-1)^2 + (y+2)^2 = 9 \] This represents a circle with center \( (1, -2) \) and radius 3. The formula for the inverse point of \( (x_1, y_1) \) with respect to a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x', y') = \left( h + \frac{r^2 (x_1 - h)}{(x_1 - h)^2 + (y_1 - k)^2}, k + \frac{r^2 (y_1 - k)}{(x_1 - h)^2 + (y_1 - k)^2} \right) \] Substituting the values \( (x_1, y_1) = (3, 2) \), \( (h, k) = (1, -2) \), and \( r = 3 \), we calculate \( (l, m) \). After simplifying, we find that \( 2l + 19m = 0 \).
Thus, the answer is \( \boxed{0} \).