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)
Question
The equation of the tangent to the circle $x^{2}+y^{2}4x+4y2=0$ 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:

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: $$ 2g = 4 \implies g = 2 $$ $$ 2f = 4 \implies f = 2 $$ $$ c = 2 $$

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 $(1, 1)$: $$ x_1 = 1, , y_1 = 1 $$

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: $$ x  3y + 2 = 0 $$

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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