First slide

Methods of solving first order first degree differential equations

Question

$The normal to a curve at p(x,y) meets the x-axis at G. If the distance of G from the origin is twice the abscisa of P, then the curve is a$

Difficult

A

parabola
B

circle
C

hyperbola
D

ellipse

Solution

$\begin{array}{l} Equation of the normal Y - y = - \frac{dx}{dy} (X - x), meets x axis \Rightarrow Y = 0 \\ - y = - \frac{dx}{dy} (X - x) \\ y \frac{d y}{d x} = X - x \Rightarrow X = x + y \frac{d y}{d x} a n d Y = 0 \\ then G (x + y \frac{dy}{dx}, 0) \Rightarrow OG = 2 x \\ \Rightarrow x + y \frac{d y}{d x} = 2 x \Rightarrow \int y d y = \int x d x \Rightarrow \frac{y^{2}}{2} = \frac{x^{2}}{2} + c \\ \Rightarrow x^{2} - y^{2} = c \end{array}$

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