The slope of tangent at any point on a curve is . If , then a value of is:

Explanation: Given that, slope of tangent at any point is which is a homogeneous differential eqn., so putting and , we get But, it is given that So, Putting

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The slope of tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2 + y^2}{2xy}, x > 0$ . If $y(2) = 0$ , then a value of $y(8)$ is:

Choose the correct answer:

Explanation

Given that, slope of tangent at any point is $\frac{x^2 + y^2}{2xy}$

\Rightarrow \frac{dy}{dx} = \frac{x^2 + y^2}{2xy}

which is a homogeneous differential eqn., so putting $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$ , we get

v + x\frac{dv}{dx} = \frac{x^2 + v^2x^2}{2x \times vx} = \frac{1 + v^2}{2v}

x\frac{dv}{dx} = \frac{1 + v^2}{2v} - v = \frac{1 + v^2 - 2v^2}{2v}

\Rightarrow x\frac{dv}{dx} = \frac{1 - v^2}{2v}

\Rightarrow x\frac{dv}{dx} = \frac{1 - v^2}{2v}

Choose the correct answer:

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