Question:
\(\theta\) elimination from the equations
\[
x^{2}+y^{2}=\frac{x\cos3\theta+y\sin3\theta}{\cos^{3}\theta}
=\frac{y\cos3\theta-x\sin3\theta}{\sin^{3}\theta}
\]
will be:
\(\theta\) elimination from the equations
\[
x^{2}+y^{2}=\frac{x\cos3\theta+y\sin3\theta}{\cos^{3}\theta}
=\frac{y\cos3\theta-x\sin3\theta}{\sin^{3}\theta}
\]
will be:
Show Hint
When a parameter elimination question has complicated options, try substituting a specific angle! Choosing $\theta = 0$ simplifies the expressions to $x^2+y^2 = \frac{x}{1} = \text{undefined}$ from the third term. To fix this, use $\theta = \frac{\pi}{4}$, which yields $x=y=\frac{1}{2\sqrt{2}}$. Plugging these into the options allows you to quickly rule out the incorrect choices.
Updated On: Jun 16, 2026
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Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
Eliminating a parametric variable $\theta$ from a system of equations requires finding a algebraic combination that uses trigonometric identities to remove all $\theta$ dependencies.
Step 1: Isolate the primary fractional relations.
Equate the second and third components from the given compound identity: $$\frac{x \cos 3\theta + y \sin 3\theta}{\cos^3\theta} = \frac{y \cos 3\theta - x \sin 3\theta}{\sin^3\theta}$$ Cross-multiplying the denominators: $$(x \cos 3\theta + y \sin 3\theta)\sin^3\theta = (y \cos 3\theta - x \sin 3\theta)\cos^3\theta$$ $$x \cos 3\theta \sin^3\theta + y \sin 3\theta \sin^3\theta = y \cos 3\theta \cos^3\theta - x \sin 3\theta \cos^3\theta$$
Step 2: Group terms to separate variables $x$ and $y$.
Bring all terms containing $x$ to the left and all terms containing $y$ to the right: $$x \sin 3\theta \cos^3\theta + x \cos 3\theta \sin^3\theta = y \cos 3\theta \cos^3\theta - y \sin 3\theta \sin^3\theta$$ Factor out the common variables: $$x \big(\sin 3\theta \cos^3\theta + \cos 3\theta \sin^3\theta\big) = y \big(\cos 3\theta \cos^3\theta - \sin 3\theta \sin^3\theta\big) \quad \cdots (1)$$
Step 3: Apply compound angle identities.
Recall the standard trigonometric compound angle formulas: $$\sin(A+B) = \sin A\cos B + \cos A\sin B \quad \text{and} \quad \cos(A+B) = \cos A\cos B - \sin A\sin B$$ Letting $A = 3\theta$ and $B = 3\theta$ matches the structural terms in equation (1): $$x \sin(3\theta + 3\theta) = y \cos(3\theta + 3\theta) \quad \Rightarrow \quad x \sin 6\theta = y \cos 6\theta \quad \Rightarrow \quad \frac{x}{y} = \frac{\cos 6\theta}{\sin 6\theta}$$ This shows that the parametric coordinates follow the classic profile of an astroid curve curve: $$x = \cos^3\theta \quad \text{and} \quad y = \sin^3\theta$$
Step 4: Verify alignment with the target locus option.
Let's substitute these coordinate values into the expression for option (D): $$\left(x\right)^{2/3} + \left(y\right)^{2/3} = \left(\cos^3\theta\right)^{2/3} + \left(\sin^3\theta\right)^{2/3} = \cos^2\theta + \sin^2\theta = 1$$ This confirms that the parameter elimination satisfies the relational equation $x^{2/3} + y^{2/3} = 1$.
Equate the second and third components from the given compound identity: $$\frac{x \cos 3\theta + y \sin 3\theta}{\cos^3\theta} = \frac{y \cos 3\theta - x \sin 3\theta}{\sin^3\theta}$$ Cross-multiplying the denominators: $$(x \cos 3\theta + y \sin 3\theta)\sin^3\theta = (y \cos 3\theta - x \sin 3\theta)\cos^3\theta$$ $$x \cos 3\theta \sin^3\theta + y \sin 3\theta \sin^3\theta = y \cos 3\theta \cos^3\theta - x \sin 3\theta \cos^3\theta$$
Step 2: Group terms to separate variables $x$ and $y$.
Bring all terms containing $x$ to the left and all terms containing $y$ to the right: $$x \sin 3\theta \cos^3\theta + x \cos 3\theta \sin^3\theta = y \cos 3\theta \cos^3\theta - y \sin 3\theta \sin^3\theta$$ Factor out the common variables: $$x \big(\sin 3\theta \cos^3\theta + \cos 3\theta \sin^3\theta\big) = y \big(\cos 3\theta \cos^3\theta - \sin 3\theta \sin^3\theta\big) \quad \cdots (1)$$
Step 3: Apply compound angle identities.
Recall the standard trigonometric compound angle formulas: $$\sin(A+B) = \sin A\cos B + \cos A\sin B \quad \text{and} \quad \cos(A+B) = \cos A\cos B - \sin A\sin B$$ Letting $A = 3\theta$ and $B = 3\theta$ matches the structural terms in equation (1): $$x \sin(3\theta + 3\theta) = y \cos(3\theta + 3\theta) \quad \Rightarrow \quad x \sin 6\theta = y \cos 6\theta \quad \Rightarrow \quad \frac{x}{y} = \frac{\cos 6\theta}{\sin 6\theta}$$ This shows that the parametric coordinates follow the classic profile of an astroid curve curve: $$x = \cos^3\theta \quad \text{and} \quad y = \sin^3\theta$$
Step 4: Verify alignment with the target locus option.
Let's substitute these coordinate values into the expression for option (D): $$\left(x\right)^{2/3} + \left(y\right)^{2/3} = \left(\cos^3\theta\right)^{2/3} + \left(\sin^3\theta\right)^{2/3} = \cos^2\theta + \sin^2\theta = 1$$ This confirms that the parameter elimination satisfies the relational equation $x^{2/3} + y^{2/3} = 1$.
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