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Vector Algebra

Question
CBSEENMA12032695
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Calculate the area bounded by the parabola y2 = 4 a x and its latus rectum.

Solution
The equation of parabola is y2 = 4 a x.    ...(1)
Let O be the vertex, S be the focus and LL' be the latus rectum of parabola.
The equation of latus rectum is x = a.
Also, we know that parabola is symmetric about x-axis.
therefore space space space required space area space space equals space 2 space left parenthesis area space OSL right parenthesis
                            equals space 2 integral subscript 0 superscript straight a straight y space dx space equals space 2 space integral subscript 0 superscript straight a 2 square root of straight a space square root of straight x space dx equals space 2 space.2 square root of straight a integral subscript 0 superscript straight a straight x to the power of 1 half end exponent dx space equals space 4 square root of straight a open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript straight a equals space 4 square root of straight a.2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript straight a space equals space fraction numerator 8 square root of straight a over denominator 3 end fraction space open square brackets straight a to the power of 3 over 2 end exponent minus 0 close square brackets equals space fraction numerator 8 square root of straight a over denominator 3 end fraction. straight a to the power of 3 over 2 end exponent space equals space 8 over 3 straight a squared space sq. space units.

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