Provide an appropriate response.
-$$\text { The curves } y = a x ^ { 2 } + b \text { and } y = 2 x ^ { 2 } + c x \text { have } a \text { common tangent line at the point } ( - 1,0 ) \text {. Find } a \text {, } b \text {, and } c \text {. }$$
A) a = 1, b = 0, c = 2
B) a - 2, b = 1, c = -1
C) a = -1, b = 1, c = -2
D) a = 1, b = -1, c = 2

Question

Quizplus · Accepted Answer

First, we need to use the fact that the two curves have a common tangent line at the point (-1,0) to find the value of c. Since the two curves have a common tangent line, they must have the same value and slope at (-1,0). Setting the two equations equal to each other and their derivatives equal to each other gives:

$$ax^2 + b = 2x^2 + cx \implies (a-2)x^2 + cx + b = 0$$

$$(2ax + c)' = (4x + c)' \implies 2ax + c = 4x + c \implies a = 2$$

Substituting this value of a into the first equation and solving for b gives:

$$2x^2 + b = y \implies 2(-1)^2 + b = 0 \implies b = -2$$

Finally, substituting a = 2 and b = -2 into the equation for the common tangent line gives:

$$y = 2x - 2$$

which is the common tangent line for the two curves at (-1,0). Therefore, the correct option is D.

Provide an Appropriate Response A) a = 1, B = 0, C = 2