Provide an appropriate response.
-$$\text { The line that is normal to the curve } x ^ { 2 } - x y + y ^ { 2 } = 36 \text { at } ( 6,6 ) \text { intersects the curve at what other point? }$$
A) (0, -6)
B) (-12, -12)
C) (-6, -6)
D) (-6, 0)

Question

Quizplus · Accepted Answer

To find the normal line to the curve at the point (6,6), we first need to find the gradient of the curve at this point. The gradient of the curve can be found by differentiating implicitly. Given $x^2 - xy + y^2 = 36$, differentiating both sides with respect to $x$ gives:$$2x - y - x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$$Solving for $\frac{dy}{dx}$ when $x = 6$ and $y = 6$:$$2(6) - 6 - 6\frac{dy}{dx} + 2(6)\frac{dy}{dx} = 0$$$$12 - 6 + 6\frac{dy}{dx} = 0$$$$6\frac{dy}{dx} = -6$$$$\frac{dy}{dx} = -1$$The gradient of the normal is the negative reciprocal of the gradient of the curve, so the gradient of the normal is $1$. The equation of the normal line can be found using the point-slope form:$$y - y_1 = m(x - x_1)$$Substituting $m = 1$, $x_1 = 6$, and $y_1 = 6$:$$y - 6 = 1(x - 6)$$$$y = x$$To find the other intersection point, substitute $y = x$ back into the original equation:$$x^2 - x^2 + x^2 = 36$$$$x^2 = 36$$$$x = \pm 6$$Since we already know the point (6,6) is on the curve, the other intersection point is $(-6, -6)$.

Provide an Appropriate Response A) (0, -6) B) (-12, -12) C) (-6, -6) D)

Provide an Appropriate Response A) (0, -6)
B) (-12, -12)
C) (-6, -6)
D)