Question

Where does the normal line to the parabola \(y = {x^2} - 1\) at the point \(\left( { - 1,0} \right)\) intersect the parabola a second time? Illustrate with a sketch.

Short answer

The point that intersects the parabola at the second time is \(\left( {\frac{3}{2},\frac{5}{4}} \right)\).

Step-by-step solution

 Normal line to a curve

The normal line to the curve \(C\) at a point \(P\) is the line through \(P\) which is perpendicular to the tangent line at \(P\).

Determine the second point of the parabola

The slope of \(f\left( x \right) = {x^2} - 1\) as shown below:

\(\begin{align}f'\left( x \right) &= \frac{d}{{dx}}\left( {{x^2} - 1} \right)\\ &= 2x - 0\\ &= 2x\end{align}\)

The slope of the tangent line at \(\left( { - 1,0} \right)\) is shown below:

\(\begin{align}f'\left( x \right) &= 2\left( { - 1} \right)\\ &= - 2\end{align}\)

The slope of the normal line is the negative reciprocal of \( - 2\), that is, \( - \frac{1}{2}\).

Determine the equation of the normal line at \(\left( { - 1,0} \right)\) as shown below:

\(\begin{align}y - 0 &= \frac{1}{2}\left( {x - \left( { - 1} \right)} \right)\\y &= \frac{1}{2}x + \frac{1}{2}\end{align}\)……(1)

Substitute \(y = \frac{1}{2}x + \frac{1}{2}\) in the equation of the parabola as shown below:

\(\begin{align}\frac{1}{2}x + \frac{1}{2} &= {x^2} - 1\\x + 1 &= 2{x^2} - 2\\2{x^2} - x - 2 - 1 &= 0\\2{x^2} - x - 3 &= 0\\\left( {2x - 3} \right)\left( {x + 1} \right) &= 0\\x &= \frac{3}{2}\,\,\,{\mathop{\rm or}\nolimits} \,\,x = - 1\end{align}\)

Substitute \(x = \frac{3}{2}\) in the equation (1) as shown below:

\(\begin{align}y &= \frac{1}{2}\left( {\frac{3}{2}} \right) + \frac{1}{2}\\ &= \frac{3}{4} + \frac{1}{2}\\ &= \frac{{3 + 2}}{4}\\ &= \frac{5}{4}\end{align}\)

Thus, the point that intersects the parabola at the second time is \(\left( {\frac{3}{2},\frac{5}{4}} \right)\).

Sketch the graph of the parabola

Sketch the graph of the parabola equation and slope of the tangent line as shown below:

It is observed from the graph that the second point of intersection in the parabola is \(\left( {\frac{3}{2},\frac{5}{4}} \right)\).