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Find the area bounded by the ellipse (x...

Find the area bounded by the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`and the ordinates `x = 0`and`x = a e`, where, `b^2=a^2(1-e^2)`and`e < 1`.

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To find the area bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and the ordinates \(x = 0\) and \(x = ae\), where \(b^2 = a^2(1 - e^2)\) and \(e < 1\), we can follow these steps: ### Step 1: Express \(y\) in terms of \(x\) From the equation of the ellipse, we can express \(y\) in terms of \(x\): \[ \frac{y^2}{b^2} = 1 - \frac{x^2}{a^2} \] This leads to: ...
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Knowledge Check

  • The area bounded by the ellipse b^(2)x^(2) + a^(2) y^(2) = a^(2) b^(2) is

    A
    `pi ab sq. units`
    B
    `2 pi "ab" sq. Units`
    C
    `pi/2 "ab" sq. units`
    D
    `3pi/2 "ab" sq. units`

  • The area bounded by the curves y= x^(2) -2x-1, e^(x) +y+1=0 and ordinates x=-1 and x= 1 is-

    A
    `(3e^(2)+2e-3)//3e`
    B
    `(e^(2)+1)//e`
    C
    `(3e^(2)-2e+3)//3`
    D
    None of these

  • If e is the eccentricity of the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 (a lt b) , then

    A
    `b^2 = a^2 (1 - e^2)`
    B
    `a^2 = b^2 (1 - e^2)`
    C
    `a^2 = b^2 (e^2 - 1)`
    D
    `b^2 = a^2 ( e^2 - 1)`