3.6 Differentiation (A-level only)

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Question 47

The path of a beam of light reflecting off a parabolic mirror is modeled by the parametric equations

2.5t^2 \text{ and } y = 5t, \quad -4 \le t \le 4

where $x\,$ and $y\,$ are measured in centimeters.

Determine the Cartesian equation of the reflective surface in the form $y^2 = f(x)$ .

[2]

bi.

A specific photon strikes the mirror at point $A$ where the parameter $t = a$ (with $\neq 1$ ). The tangent to the curve at point $A$ makes an angle $θ\theta$ with a line through $A$ parallel to the $x$ -axis. The point $B$ is located at $(2.5, 0)$ . The line segment $A B$ makes an angle $ϕ\phi$ with the positive $x$ -axis.

By calculating the gradient of the curve, show that $tan⁡θ=1a\tan \theta = \frac{1}{a}$ .

[3]

bii.

Find an expression for $tan⁡ϕ\tan \phi$ in terms of $a$ , simplifying your answer.

[3]

biii.

Hence, show that $tan⁡2θ=tan⁡ϕ\tan 2\theta = \tan \phi$ .

[3]

3.6 Differentiation (A-level only)

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Question 47

The path of a beam of light reflecting off a parabolic mirror is modeled by the parametric equations

2.5t^2 \text{ and } y = 5t, \quad -4 \le t \le 4

where $x\,$ and $y\,$ are measured in centimeters.

Determine the Cartesian equation of the reflective surface in the form $y^2 = f(x)$ .

[2]

bi.

By calculating the gradient of the curve, show that $tan⁡θ=1a\tan \theta = \frac{1}{a}$ .

[3]

bii.

Find an expression for $tan⁡ϕ\tan \phi$ in terms of $a$ , simplifying your answer.

[3]

biii.

Hence, show that $tan⁡2θ=tan⁡ϕ\tan 2\theta = \tan \phi$ .

[3]

3.6 Differentiation (A-level only)

Question 47

Markscheme

3.6 Differentiation (A-level only)

Question 47

Markscheme

3.6 Differentiation (A-level only)

Question 47

3.6 Differentiation (A-level only) Questions

3.6 Differentiation (A-level only)

Question 47

3.6 Differentiation (A-level only) Questions