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Maths-

General

Easy

Question

The differential equation of all circles which pass through the origin and whose center lie on y-axis is

$(x^{2} - y^{2}) \frac{d y}{d x} - 2 x y = 0$
$(x^{2} - y^{2}) \frac{d y}{d x} + 2 x y = 0$
$(x^{2} - y^{2}) \frac{d y}{d x} - x y = 0$
$(x^{2} - y^{2}) \frac{d y}{d x} + x y = 0$

hint Hint:

In the given question we have to find the differential equation of the equation of the circle center lying on y-axis for this we will write the equation of the circle then we will simplify the equation and further differentiate it to find the value of a. Later substitute it in the equation given to find the required equation.

The correct answer is: $open parentheses x squared minus y squared close parentheses fraction numerator d y over denominator d x end fraction minus 2 x y equals 0$

$c e n t e r open parentheses 0 comma a close parentheses E q u a t i o n space o f space c i c l e comma space open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared rightwards double arrow x squared plus open parentheses y minus a close parentheses squared equals a squared rightwards double arrow x squared plus y squared plus a squared minus 2 y a equals a squared rightwards double arrow x squared plus y squared equals 2 y a...... space e q u space 1 d i f f e r e n t a i t i n g rightwards double arrow fraction numerator d over denominator d x end fraction x squared plus fraction numerator d over denominator d x end fraction y squared equals 2 fraction numerator d over denominator d x end fraction y a rightwards double arrow 2 x plus 2 y fraction numerator d y over denominator d x end fraction equals 2 a fraction numerator d y over denominator d x end fraction rightwards double arrow a equals fraction numerator x plus y begin display style fraction numerator d y over denominator d x end fraction end style over denominator begin display style fraction numerator d y over denominator d x end fraction end style end fraction........ space e q u space 2 F r o m space e q u space 1 space a n d space 2 x squared plus y squared equals 2 y open parentheses fraction numerator x plus y begin display style fraction numerator d y over denominator d x end fraction end style over denominator begin display style fraction numerator d y over denominator d x end fraction end style end fraction close parentheses rightwards double arrow x squared fraction numerator d y over denominator d x end fraction plus y squared fraction numerator d y over denominator d x end fraction equals 2 x y plus 2 y squared fraction numerator d y over denominator d x end fraction rightwards double arrow open parentheses x squared minus y squared close parentheses fraction numerator d y over denominator d x end fraction equals 2 x y rightwards double arrow open parentheses x squared minus y squared close parentheses fraction numerator d y over denominator d x end fraction minus 2 x y equals 0$

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