First slide

Methods of solving first order first degree differential equations

Question

$If the eccentricity of curve for which tangent at point P intersects the y axis at M above x -axis such$
$that origin is equidistant from M and point of tangency, is e, then value of 6 e^{2} is$

Difficult

Solution

We have
$\begin{array}{l} OP = OM \\ \Rightarrow y - x \frac{d y}{d x} = \sqrt{x^{2} + y^{2}} \\ \Rightarrow \frac{d y}{d x} = \frac{y - \sqrt{x^{2} + y^{2}}}{x} \\ \Rightarrow \frac{d y}{d x} = \frac{y}{x} - \sqrt{1 + {(\frac{y}{x})}^{2}} \\ putting \frac{y}{x} = v \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx} \\ \Rightarrow v + x \frac{dv}{dx} = v - \sqrt{1 + v^{2}} \\ \Rightarrow \log (v + \sqrt{1 + v^{2}}) = \log \frac{c}{x} \\ \Rightarrow (v + \sqrt{1 + v^{2}}) = \frac{c}{x} \\ \Rightarrow y + \sqrt{x^{2} + y^{2}} = c \\ ⇒curve is parabola \\ ∴ e = 1 \end{array}$

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