Solve the problem.
-$$\text { Write an equation for the tangent line to the curve } x ^ { 2 } - 10 x y + y ^ { 2 } = 12 \text { at the point } ( - 1,1 ) \text {. }$$
A) y = x + 1
B) y = x - 2
C) x - y + 2 = 0
D) x + y = 1

Question

Quizplus · Accepted Answer

First, we need to find the derivative of the curve at the given point to get the slope of the tangent line. We can find the derivative implicitly by differentiating both sides of the equation with respect to x:

$2x - 10y - 10x\frac{dy}{dx} + 2y \frac{dy}{dx} = 0$

Simplifying and solving for $\frac{dy}{dx}$ gives:

$\frac{dy}{dx} = \frac{10y-2x}{2y-10x}$

Plugging in the point (-1,1) gives:

$\frac{dy}{dx} = \frac{10(1)-2(-1)}{2(1)-10(-1)} = -\frac{3}{4}$

So the slope of the tangent line is $-\frac{3}{4}$.

We can now use the point-slope form of a line to find the equation of the tangent line:

$y-1 = -\frac{3}{4}(x+1)$

Simplifying gives:

$y = -\frac{3}{4}x - \frac{1}{4}$

Rearranging and adding x to both sides gives:

$x - y + 2 = 0$.

Therefore, the equation of the tangent line is $x - y + 2 = 0$, which is choice C.