Provide an appropriate response.
-At the two points where the curve $x ^ { 2 } + 2 x y + y ^ { 2 } = 9$ crosses the $x$-axis, the tangents to the curve are parallel. What is the common slope of these tangents?
A) 1
B) $- 3$
C) $- 1$
D) $\frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

First, we need to find the points where the curve crosses the x-axis. This means that y = 0, so we can substitute that into the equation:
$$x^2 + 2xy + y^2 = 9$$
$$x^2 + 0 + 0 = 9$$
$$x = \pm 3$$
So the two points are (-3, 0) and (3, 0).
Next, we need to find the slopes of the tangents to the curve at these two points. We can use implicit differentiation to find the slope of the curve at any point (x, y):
$$x^2 + 2xy + y^2 = 9$$
$$2x + 2y \frac{dy}{dx} + 2y + 2x \frac{dy}{dx} = 0$$
$$(2x + 2y) \frac{dy}{dx} = -2x - 2y$$
$$\frac{dy}{dx} = -\frac{x+y}{x+y}$$
$$\frac{dy}{dx} = -1$$
So the slope of the curve at any point is -1. Since the tangents at the two x-intercepts are parallel, they must have the same slope of -1, so the answer is C.

Provide an Appropriate Response x2+2xy+y2=9x ^ { 2 } + 2 x y + y ^ { 2 } = 9x2+2xy+y2=9

Provide an Appropriate Response $x ^ { 2 } + 2 x y + y ^ { 2 } = 9$