If axcosθ+bysinθ=1,axsinθ−bycosθ=1, then eliminate θ

Given, axcos θ+bysin θ=1,axsin θ−bycos θ=1Squaring and adding them ⇒a2x2cos2θ +b2y2sin2θ +ab2xysinθ cosθ + a2x2sin2θ +b2y2cos2θ −ab2xysinθ cosθ =1+1 ⇒a2x2+b2y2=2

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If {x}{a} +{y}{b} =1,{x}{a} -{y}{b} =1, then eliminate

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If $\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1, \frac{x}{a} sin θ - \frac{y}{b} cos θ = 1,$ then eliminate $θ$

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 2

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 2

\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{a ^{2}} = 2

\frac{x ^{2}}{b ^{2}} - \frac{y ^{2}}{a ^{2}} = 2

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\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1, \frac{x}{a} sin θ - \frac{y}{b} cos θ = 1,

then eliminate

θ

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Text solutionVerified

Given,

\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1, \frac{x}{a} sin θ - \frac{y}{b} cos θ = 1

Squaring and adding them

\Rightarrow \frac{x ^{2}}{a ^{2}} cos^{2} θ + \frac{y ^{2}}{b ^{2}} sin^{2} θ + \frac{2 x y}{ab} sin θ cos θ +

\frac{x ^{2}}{a ^{2}} sin^{2} θ + \frac{y ^{2}}{b ^{2}} cos^{2} θ - \frac{2 x y}{ab} sin θ cos θ = 1 + 1

\Rightarrow \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 2

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Question Text	If $\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1, \frac{x}{a} sin θ - \frac{y}{b} cos θ = 1,$ then eliminate $θ$
Answer Type	Text solution:1
Upvotes	150

If ax​cosθ+by​sinθ=1,ax​sinθ−by​cosθ=1, then eliminate θ

Text solutionVerified

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If $\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1, \frac{x}{a} sin θ - \frac{y}{b} cos θ = 1,$ then eliminate $θ$