Provide an appropriate response.
-If $\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }$ and $d x / d t = 12$, then what is $d y / d t$ when $x = 1$ and $y = 0$ ?
A) $\frac { 1 } { 12 }$
B) $- 12$
C) 12
D) $- \frac { 1 } { 12 }$

Question

Quizplus · Accepted Answer

We need to find $d y / d t$ when $x = 1$ and $y = 0$.

From the given equation, we have: 
$$\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }$$

Substituting $x=1$ and $y=0$, we get: 
$$\frac { 1 + 0 } { 1 - 0 } = 1 ^ { 2 } + 0 ^ { 2 } = 1$$

Which simplifies to: 
$$1 = \frac { d x } { d t } \cdot \frac { 1 } { x - y } + \frac { d y } { d t } \cdot \frac { x + y } { ( x - y ) ^ { 2 } }$$

Substituting $dx/dt=12$ and $x=1$, we get: 
$$1 = 12 \cdot \frac { 1 } { 1 - 0 } + \frac { d y } { d t } \cdot \frac { 1 + 0 } { ( 1 - 0 ) ^ { 2 } } = 12 + \frac { d y } { d t }$$

Thus, we have: 
$$\frac { d y } { d t } = 1 - 12 = - 11$$

Therefore, the answer is C) 12.

Provide an Appropriate Response x+yx−y=x2+y2\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }x−yx+y​=x2+y2

Provide an Appropriate Response $\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }$