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Vector Algebra

Question
CBSEENMA12032714

Find the area of the region enclosed by the parabola y2 = 4 a x and the line y = mx.

Solution
The equation of parabola is y2 = 4 a x    ... (1)
The line y = mx meets (1) where straight m squared straight x squared space equals space 4 space straight a space straight x
or
   straight x space left parenthesis straight m squared straight x space minus space 4 straight a right parenthesis space equals space 0 space space or space straight x space equals space 0 comma space space fraction numerator 4 straight a over denominator straight m squared end fraction
therefore space space straight y space equals space 0 comma space space fraction numerator 4 straight a over denominator straight m end fraction
rightwards double arrow space space space straight O space is space left parenthesis 0 comma space 0 right parenthesis space space and space straight A comma space is space open parentheses fraction numerator 4 straight a over denominator straight m squared end fraction comma space fraction numerator 4 straight a over denominator straight m end fraction close parentheses
Area bounded by curve (1) above the x - axis ordinates straight x space equals space 0 comma space space straight x space equals space fraction numerator 4 straight a over denominator straight m squared end fraction is
equals space integral subscript 0 superscript 4 straight a divided by straight m squared end superscript straight y space dx space equals space integral subscript 0 superscript 4 straight a divided by straight m squared end superscript square root of straight x space dx space equals 2 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 fraction numerator 4 straight a over denominator straight m squared end fraction end superscript
equals space fraction numerator 4 square root of straight a over denominator 3 end fraction open square brackets open parentheses fraction numerator 4 straight a over denominator straight m squared end fraction close parentheses to the power of 3 over 2 end exponent minus 0 close square brackets space space equals space fraction numerator 4 square root of straight a over denominator 3 end fraction cross times fraction numerator 8 straight a to the power of begin display style 3 over 2 end style end exponent over denominator straight m cubed end fraction space equals space 32 over 3 straight a squared over straight m cubed
Area bounded by line y = m x, above the x-axis and ordinates x = 0,  straight x space equals space fraction numerator 4 straight a over denominator straight m squared end fraction space is
                     equals space integral subscript 0 superscript 4 straight a divided by straight m squared end superscript mx space dx space equals space straight m open square brackets straight x squared over 2 close square brackets subscript 0 superscript fraction numerator 4 straight a over denominator straight m squared end fraction end superscript space equals space straight m over 2 open square brackets straight x squared close square brackets subscript 0 superscript fraction numerator 4 straight a over denominator straight m squared end fraction end superscript
equals space straight m over 2 open square brackets fraction numerator 16 straight a squared over denominator straight m to the power of 4 end fraction minus 0 close square brackets space equals space straight m over 2 cross times fraction numerator 16 straight a squared over denominator straight m to the power of 4 end fraction space equals space fraction numerator 8 straight a squared over denominator straight m cubed end fraction space sq. space units
therefore space space space required space area space equals space fraction numerator 32 space straight a squared over denominator 3 space straight m cubed end fraction minus fraction numerator 8 space straight a squared over denominator straight m cubed end fraction space equals space fraction numerator 32 straight a squared minus 24 straight a squared over denominator 3 straight m cubed end fraction space equals space fraction numerator 8 straight a squared over denominator 3 straight m cubed end fraction space sq. space units.

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