Question

The equation of the tangent to the circle $x^{2}+y^{2}-4x+4y-2=0$ at $(1,1)$ is
A. $2x+3y=13
B. $3x+2y=13
C. $2x+3y=\sqrt{13}
D. $3x+2y=\sqrt{13}

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Answer

To determine the equation of the tangent to the circle $x^2 + y^2 - 4x + 4y - 2 = 0$ at the point $(1, 1)$, we can follow these steps:

  1. Rewrite the Circle's Equation in Standard Form: The given equation is: $$ x^2 + y^2 - 4x + 4y - 2 = 0 $$ For a circle, the general form is: $$ x^2 + y^2 + 2gx + 2fy + c = 0 $$ By comparing: $$ 2g = -4 \implies g = -2 $$ $$ 2f = 4 \implies f = 2 $$ $$ c = -2 $$

  2. Equation of the Tangent: The equation of the tangent to the circle at a point $(x_1, y_1)$ is given by: $$ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 $$ For the point $(1, 1)$: $$ x_1 = 1, , y_1 = 1 $$

  3. Substitute into the Tangent Equation: Using the values $g = -2$, $f = 2$, and $c = -2$: $$ x(1) + y(1) + (-2)(x + 1) + (2)(y + 1) - 2 = 0 $$ Simplifying: $$ x + y - 2(x + 1) + 2(y + 1) - 2 = 0 $$ $$ x + y - 2x - 2 + 2y + 2 - 2 = 0 $$ Combine like terms: $$ -x + 3y - 2 = 0 $$ Multiply through by $-1$ to simplify: $$ x - 3y + 2 = 0 $$

  4. Match with Given Options: Comparing with the provided options, the correct answer matches the first option.

Therefore, the equation of the tangent is $ \boxed{x - 3y + 2 = 0}$.


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