The value of ‘b’ such that the solution of x=by dydx represents a family of circles with centre at origin is

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The value of ‘b’ such that the solution of $x = by \frac{dy}{dx}$ represents a family of circles with centre at origin is

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b = –1

for all value of b

for any value of b the solution does not represents the circle

b = 1

answer is B.

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Detailed Solution

$\begin{array}{l} x = b y \frac{d y}{d x} \\ \Rightarrow x d x = b y d y Integrating on both sides \begin{aligned} \Rightarrow \frac{x^{2}}{2} = \frac{b y^{2}}{2} + c \\ x^{2} = b y^{2} + 2 c \to (1) \end{aligned} eq (1) comparing with x^{2} + y^{2} = r^{2} \Rightarrow b = - 1 \end{array}$

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