Equations x - acos- and y- bsin- represent a conic section whose eccentricity e is given by

The correct option is B e^2 = a^2 b^2a^2 Given equation can be rewritten as cosθ = xa and sinθ = yb ∵cos^2θ +sin^2θ = 1 ∴ (xa )^2 + ...

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Standard XII

Mathematics

Eccentricity

Equations x...

Question

Equations $x = a cos θ$ and $y = b sin θ$ represent a conic section whose eccentricity $e$ is given by

e^{2} = \frac{a^{2} + b^{2}}{a^{2}}

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e^{2} = \frac{a^{2} + b^{2}}{b^{2}}

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e^{2} = \frac{a^{2} - b^{2}}{a^{2}}

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None of these

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Solution

The correct option is B $e^{2} = \frac{a^{2} - b^{2}}{a^{2}}$
Given equation can be rewritten as
$cos θ = \frac{x}{a}$ and $sin θ = \frac{y}{b}$
$∵ {cos}^{2} θ + {sin}^{2} θ = 1$
$∴ {(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1$
Thus, it represents an equation of an ellipse.
$∴ e = \sqrt{1 - \frac{b^{2}}{a^{2}}} \Rightarrow e^{2} = \frac{a^{2} - b^{2}}{a^{2}}$ .

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Equations x=acosθ and y=bsinθ represent a conic section whose eccentricity e is given by

The correct option is B e2=a2−b2a2Given equation can be rewritten ascosθ=xa and sinθ=yb∵cos2θ+sin2θ=1∴(xa)2+(yb)2=1Thus, it represents an equation of an ellipse.∴e=√1−b2a2⇒e2=a2−b2a2.

Equations $x = a cos θ$ and $y = b sin θ$ represent a conic section whose eccentricity $e$ is given by