(See the Guided Projects for additional applications of conic sections.)

Reflection property of parabolas Consider the parabola \(y=x^{2} /(4 p)\) with its focus at F(0, p) (see figure). We know that light is reflected from a surface in such a way that the angle of incidence equals the angle of reflection. The goal is to show that the angle between the ray \(\ell\) and the tangent line L (\(\alpha\) in the figure) equals the angle between the line PF and L (\(\beta\) in the figure). If these two angles are equal, then the reflection property is proved because \(\ell\) is reflected through F.

a. Let \(P\left(x_{0}, y_{0}\right)\) be a point on the parabola. Show that the slope of the line tangent to the curve at P is \(\tan \theta=x_{0} /(2 p)\).

b. Show that \(\tan \varphi=\left(p-y_{0}\right) / x_{0}\).

c. Show that \(\alpha=\pi / 2-\theta\); therefore, \(\tan \alpha=\cot \theta\).

d. Note that \(\beta=\theta+\varphi\). Use the tangent addition formula

\(\tan (\theta+\varphi)=\frac{\tan \theta+\tan \varphi}{1-\tan \theta \tan \varphi}\) to show that \(\tan \alpha=\tan \beta=2 p / x_{0}\).

e. Conclude that because \(\alpha\) and \(\beta\) are acute angles, \(\alpha=\beta\).

Solution 83EStep 1 of 5:a) In this problem we need to Show that the slope of the line tangent to the curve at P is tan(.Given parabola is y = with its focus at F (0 , p).Let p() be a point on the parabola.Let be an inclination of the tangent line , then the slope of the tangent line is tan(We know that ,slope is denoted by = m = tan(Consider , y = then differentiate both sides with respect to x we get , = () = , since cwhere c is constant. = .Therefore , the slope of the tangent line at the point p() is : = m = tan(=