Question

For what values of \(a\) and \(b\) is the line \(2x + y = b\) tangent to the parabola \(y = a{x^2}\) when \(x = 2\)?

Short answer

The values of \(a = - \frac{1}{2}\) and \(b = 2\).

Step-by-step solution

Tangent to a parabola

A tangent to a parabola is a straight line that passes through the point lying on the parabola just by touching it at that particular point without intersecting the curve.

Evaluating the equation of a parabola.

Thegiven equation is \(y = a{x^2}\).

Differentiating with respect to\(x\), we have:

\(y' = \;2ax\)

Now, the equation of a tangent line is \(2x + y = b\) has slope \(2\), so at \(x = 2\) we have:

\(\begin{aligned}y' &= - 2\\2a\left( 2 \right) &= - 2\\a &= - \frac{1}{2}\end{aligned}\)

Putting in the equation of parabola at\(x = 2\):

\(\begin{aligned}y &= a{x^2}\\ &= - \frac{1}{2}{\left( 2 \right)^2}\\ &= - 2\end{aligned}\)

Evaluating the tangent equation.

Now, for the tangent line \(2x + y = b\)at point\(\left( {2, - 2} \right)\), we have:

\(\begin{aligned}2x + y &= b\\4 - 2 &= b\\b &= 2\end{aligned}\)

So, the obtained values are:

\(\begin{aligned}a &= - \frac{1}{2}\\b &= 2\end{aligned}\)

Hence, this is the required answer.