Exams Joint Entrance Examination Mathematics Differential Equations Question 38 Question 38 of 490 Medium MCQ
Suppose the solution of the differential equation d y d x = ( 2 + α ) x − β y + 2 β x − 2 α y − ( β γ − 4 α ) \frac{dy}{dx}=\frac{(2+\alpha )x-\beta y+2}{\beta x-2\alpha y-(\beta \gamma -4\alpha )} d x d y = β x − 2 α y − ( β γ − 4 α ) ( 2 + α ) x − β y + 2 represents a circle passing through origin. Then the radius of this circle is :
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d y d x = ( 2 + α ) x − β y + 2 β x − 2 α y − ( β γ − 4 α ) β x d y − 2 α y d y − ( β γ − 4 α ) d y = 2 x d x + α x d x − β y d x + 2 d x β ∫ ( x d y + y d y ) − α y 2 − ( β γ − 4 x ) y = x 2 + α x 2 2 + 2 x β x y − α y 2 − ( β γ − 4 α ) y = x 2 + α x 2 2 + 2 x ( 1 + α 2 ) x 2 + α y 2 − β x y + 2 x + ( β γ − 4 α ) y = 0 ∵ ⇒ β = 01 + α 2 = α ⇒ α = 2 \begin{gathered} \frac{dy}{dx}=\frac{(2+\alpha )x-\beta y+2}{\beta x-2\alpha y-(\beta \gamma -4\alpha )} \\ \beta xdy-2\alpha ydy-(\beta \gamma -4\alpha )dy \\ =2xdx+\alpha xdx-\beta ydx+2dx \\ \beta \int (xdy+ydy)-\alpha {y}^{2}-(\beta \gamma -4x)y={x}^{2}+\frac{\alpha {x}^{2}}{2}+2x \\ \beta xy-\alpha {y}^{2}-(\beta \gamma -4\alpha )y={x}^{2}+\frac{\alpha {x}^{2}}{2}+2x \\ (1+\frac{\alpha }{2}){x}^{2}+\alpha {y}^{2}-\beta xy+2x+(\beta \gamma -4\alpha )y=0 \\ \because \\ \Rightarrow \beta =01+\frac{\alpha }{2}=\alpha \\ \Rightarrow \alpha =2 \end{gathered} d x d y = β x − 2 α y − ( β γ − 4 α ) ( 2 + α ) x − β y + 2 β x d y − 2 α y d y − ( β γ − 4 α ) d y = 2 x d x + α x d x − β y d x + 2 d x β ∫ ( x d y + y d y ) − α y 2 − ( β γ − 4 x ) y = x 2 + 2 α x 2 + 2 x β x y − α y 2 − ( β γ − 4 α ) y = x 2 + 2 α x 2 + 2 x ( 1 + 2 α ) x 2 + α y 2 − β x y + 2 x + ( β γ − 4 α ) y = 0 ∵ ⇒ β = 01 + 2 α = α ⇒ α = 2
∴ C : 2 x 2 + 2 y 2 + 2 x − 8 y = 0 x 2 + y 2 + x − 4 y = 0 = 1 4 + 4 − 0 = 17 2 \begin{gathered} \therefore C:2{x}^{2}+2{y}^{2}+2x-8y=0 \\ {x}^{2}+{y}^{2}+x-4y=0 \\ =\sqrt{\frac{1}{4}+4-0} \\ \, =\frac{\sqrt{17}}{2} \end{gathered} ∴ C : 2 x 2 + 2 y 2 + 2 x − 8 y = 0 x 2 + y 2 + x − 4 y = 0 = 4 1 + 4 − 0 = 2 17 Answer the question to unlock