Question

Find the area bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space the space$ ordinates x = a e and x = 0 where b² = a² (1 - e²) and e < 1.

Solution

The equation of ellipse
$straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1$
or $straight y squared over straight b squared space equals space 1 minus straight x squared over straight a squared$
or $straight y squared space equals space straight b squared over straight a squared left parenthesis straight a squared minus straight x squared right parenthesis$
or $straight y space equals space straight b over straight a square root of straight a squared minus straight x squared end root$ (in the first quadrant)
Ordinates are x = 0, x = a e.
Required area = $2 space integral subscript 0 superscript straight a space straight e end superscript space straight y space dx$ [ $because$ ellipse is symmetrical about x-axis]
$equals space fraction numerator 2 straight b over denominator straight a end fraction integral subscript 0 superscript ae square root of straight a squared minus straight x squared end root dx space equals space fraction numerator 2 straight b over denominator straight a end fraction open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript 0 superscript straight a space straight e end superscript equals space fraction numerator 2 straight b over denominator straight a end fraction open square brackets open curly brackets fraction numerator ae square root of straight a squared minus straight a squared straight e squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent open parentheses ae over straight a close parentheses close curly brackets minus open curly brackets 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close curly brackets close square brackets equals space straight b over straight a open square brackets ae. straight a square root of 1 minus straight e squared end root plus straight a squared sin to the power of negative 1 end exponent left parenthesis straight e right parenthesis close square brackets space equals space ab space open parentheses straight e square root of 1 minus straight e squared end root plus sin to the power of negative 1 end exponent straight e close parentheses$

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Find the area bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space the space$ ordinates x = a e and x = 0 where b² = a² (1 - e²) and e < 1.

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Find the area bounded by the ellipse ordinates x = a e and x = 0 where b2 = a2 (1 - e2) and e < 1.

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the area bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space the space$ ordinates x = a e and x = 0 where b² = a² (1 - e²) and e < 1.