Question

for what values of a and b is the line 5x + y = b tangent to the curve y = ax^2 when x = 2?

Explanation:

Step1: Find the derivative of the curve

The curve is $y = ax^{2}$. Using the power - rule for differentiation $\frac{d}{dx}(x^{n})=nx^{n - 1}$, the derivative $y^\prime=2ax$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $y^\prime$. We get $y^\prime|_{x = 2}=4a$.

Step3: Rewrite the line equation in slope - intercept form

The line is $5x + y=b$, which can be rewritten as $y=-5x + b$. The slope of this line is $m=-5$.

Step4: Set the slope of the line equal to the slope of the curve

Since the line is tangent to the curve, the slopes are equal. So $4a=-5$, then $a=-\frac{5}{4}$.

Step5: Find the $y$ - value on the curve at $x = 2$

Substitute $x = 2$ and $a = -\frac{5}{4}$ into the curve equation $y = ax^{2}$. So $y=-\frac{5}{4}\times2^{2}=-5$.

Step6: Substitute $x = 2$, $y=-5$ into the line equation

Substitute $x = 2$ and $y=-5$ into the line equation $5x + y=b$. We have $5\times2+( - 5)=b$, which simplifies to $b = 5$.

Answer:

$a=-\frac{5}{4}$, $b = 5$