Question

What is the value of \(c\) such that the line \(y = 2x + 3\) is tangent to the curve \(y = c{x^2}\).

Short answer

The required value is \(c = - \frac{1}{3}\).

Step-by-step solution

Tangent to a Parabola

A tangent to a parabola can be defined as a straight-line curve that passes through the only point on the parabola which can be expressible in the form of an equation of the line.

Evaluating the equation of a parabola

Thegiven equation is:

\(y = c{x^2}\)

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

\(y' = \;2cx\)

Now, the equation of a tangent line is \(y = 2x + 3\) has slope 2, so at \(x = a\), we have:

\(\begin{aligned}y'\left| {_{x = a}} \right. &= 2\\2ca &= 2\\c &= \frac{1}{a}\end{aligned}\)

Evaluating the tangent equation.

Now, the tangent equation at \(x = a\)will be:

\(\begin{aligned}c{a^2} &= 2a + 3\\a &= 2a + 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{as}}\,\,c = \frac{1}{a}} \right)\\a &= - 3\end{aligned}\)

So, we have:

\(\begin{aligned}c &= \frac{1}{a}\\ &= - \frac{1}{3}\end{aligned}\)

Hence, the required value is \(c = - \frac{1}{3}\).