Question

x = 2 cos theta - 3 sin theta , y = cos theta + 2 sin theta , eliminate theta.

Answer 1

Answer :

Given that,

x = 2 cosθ - 3 sinθ ...(i)

y = cosθ + 2 sinθ

⇒ 2y = 2 cosθ + 4 sinθ ...(ii)

Now, (ii) - (i) ⇒

2y - x = 2 cosθ + 4 sinθ - 2 cosθ + 3 sinθ

⇒ 2y - x = 7 sinθ ...(iii)

Again, (ii) × 7 ⇒

7y = 7 cosθ + 2 (7 sinθ)

⇒ 7y = 7 cosθ + 2 (2y - x), by (iii)

⇒ 7 cosθ = 3y + 2x ...(iv)

Finally, we have

cosθ = $\frac{\text{3y+2x}}{7}$

sinθ = $\frac{\text{2y-x}}{7}$

We know that,

sin²θ + cos²θ = 1

$\implies {(\frac{2y - x}{7})}^{2} + {(\frac{3y + 2x}{7}})^{2} = 1$

$\implies \frac{4{y}^{2} - 4xy + {x}^{2}}{49} + \frac{9{y}^{2} + 12xy + 4{x}^{2}}{49} = 1$

$\implies \frac{4{y}^{2} - 4xy + {x}^{2} + 9{y}^{2} + 12xy + 4{x}^{2}}{49}=1$

⇒ 5x² + 13y² + 8xy = 49,

which is the required equation.

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Answer 2

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