Trigonometry & Inverse Trigonometry
Elimination of Trigonometric Parameter
Grade 11
Question:
<p><strong>Ex. 84:</strong> If <i>x</i> sin³θ + <i>y</i> cos³θ = sin θ cos θ and <i>x</i> sin θ - <i>y</i> cos θ = 0, then (<i>x</i>, <i>y</i>) lie on</p>
<p>(a) a circle</p>
<p>(b) a parabola</p>
Step-by-Step Solution
Key Concept: Use the linear constraint to eliminate the parameter θ and determine the geometric curve on which the points (x, y) lie.
<p><strong>Step 1:</strong> From the second equation: <i>x</i> sin θ = <i>y</i> cos θ</p><p><strong>Step 2:</strong> Substitute into the first equation: <i>x</i> sin³θ + <i>y</i> cos³θ = sin θ cos θ</p><p><strong>Step 3:</strong> Using the relation from the second equation to eliminate θ and solve for the locus of (<i>x</i>, <i>y</i>)</p><p><strong>Step 4:</strong> The locus is a circle.</p>
Correct Answer: A