Find the area of the smaller part of the circle x 2 + y 2 = a cut oil by the line - Wired Faculty

Question

Find the area of the smaller part of the circle x² + y² = a cut oil by the line $straight x equals fraction numerator straight a over denominator square root of 2 end fraction$

Solution

The equation of circle is
$straight x squared minus straight y squared space equals straight a squared$ ...(1)
The equation of line is
$straight x space equals space fraction numerator straight a over denominator square root of 2 end fraction$ ...(2)
From (1) and (2), we get,
$straight a squared over 2 minus straight y squared space equals space straight a squared space space space space or space space space straight y squared space equals space straight a squared over 2$
$therefore space space space space space straight y space equals space fraction numerator straight a over denominator square root of 2 end fraction$
$because space space space straight C$ is $open parentheses fraction numerator straight a over denominator square root of 2 end fraction comma space fraction numerator straight a over denominator square root of 2 end fraction close parentheses$

$equals space 2 integral subscript fraction numerator straight a over denominator square root of 2 end fraction end subscript superscript straight a straight y space dx space equals space 2 integral subscript fraction numerator straight a over denominator square root of 2 end fraction end subscript superscript straight a square root of straight a squared minus straight x squared end root dx$
$equals space 2 open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript fraction numerator straight a over denominator square root of 2 end fraction end subscript superscript straight a equals space 2 open parentheses fraction numerator straight a square root of straight a squared minus straight a squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent 1 close parentheses space minus space 2 open parentheses fraction numerator straight a over denominator 2 square root of 2 end fraction square root of straight a squared minus straight a squared over 2 end root plus straight a squared over 2 sin to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 2 end fraction close parentheses equals space 2 open parentheses 0 plus straight a squared over 2 cross times straight pi over 2 close parentheses minus 2 open parentheses fraction numerator straight a over denominator 2 square root of 2 end fraction cross times fraction numerator straight a over denominator square root of 2 end fraction plus straight a squared over 2 cross times straight pi over 4 close parentheses equals space πa squared over 2 minus straight a squared over 2 minus πa squared over 4 space equals space πa squared over 4 minus straight a squared over 2$
$equals space straight a squared over 4 left parenthesis straight pi minus 2 right parenthesis space sq. space units$

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Find the area of the smaller part of the circle x² + y² = a cut oil by the line $straight x equals fraction numerator straight a over denominator square root of 2 end fraction$

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Vector Algebra

Find the area of the smaller part of the circle x2 + y2 = a cut oil by the line

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the area of the smaller part of the circle x² + y² = a cut oil by the line $straight x equals fraction numerator straight a over denominator square root of 2 end fraction$