Question

If $x = h + a cos θ$ , $y = k + b sin θ$ , then prove that:
$(\frac{x - h}{a})^{2} + (\frac{y - k}{b})^{2} = 1$

Approach

Given parametric equations $x = h + a cos θ$ and $y = k + b sin θ$ , prove $(\frac{x - h}{a})^{2} + (\frac{y - k}{b})^{2} = 1$ . Rearrange to get $cos θ = \frac{x - h}{a}$ and $sin θ = \frac{y - k}{b}$ . Use $cos^{2} θ + sin^{2} θ = 1$ to obtain the result.

Step-by-step working

1
$x = h + a cos θ \Rightarrow cos θ = \frac{x - h}{a}$
1 mark
2
$y = k + b sin θ \Rightarrow sin θ = \frac{y - k}{b}$
1 mark
3
Using $cos^{2} θ + sin^{2} θ = 1$ : $(\frac{x - h}{a})^{2} + (\frac{y - k}{b})^{2} = 1$
1 mark

Concepts used

trigonometric identityproveparametric equationstrigonometric simplification

Approach

Step-by-step working

Concepts used

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