Question

Find the shape of a mirror which has the property that rays from a point 0 on the axis are reflected into a parallel beam. Hint: Take the point 0 at the origin. Show from the figure that $\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{=}\mathbf{2}\frac{\mathbf{y}}{\mathbf{x}}$. Use the formula for $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }\mathbf{2}$to express this in terms of $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }\mathbf{=}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}$and solve the resulting differential equation. (Hint: See Problem 16.)

Short answer

Answer

The shape of the mirror is given by the equations $y=-e^{\frac{c_1}{2}}\sqrt{e^{c_1}-2x}$and $y=e^{\frac{c_1}{2}}\sqrt{e^{c_1}-2x}$.

Step-by-step solution

 Given information

The rays from the point 0 on the axis are reflected into the parallel beam as shown below.

Trigonometry formula

$\mathit{t}\mathit{a}\mathit{n}\mathbf{2}\mathit{\theta }\mathbf{=}\frac{\mathbf{2}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }}{\mathbf{1}\mathbf{-}\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}$.

The slope of reflected rays

The angle between the tangent of the point $x,y)$and the incident ray is $\theta$. So the angle between the tangent and extend the reflected beam with the x-axis is $2\theta$. So,

role="math" localid="1655278334064" $\tan 2\theta ·=\frac{y}{x}$

Also, for the slope of the reflected ray:

$\tan \theta =\frac{dy}{dx}$

Recall that $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$.

Then,

Find the differential equation

Now, usein the equation to get the differential equation as:

The solution of the differential equation can be given as:

That means, the shape of the mirror is given by the equationsand.