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Q.

$The tangent at any point P on the curve meets the y-axis at N and M is the foot of the$ $perpendicular from P on the x - axis . If the$ $area of Δ O M N = 1 the curve is$

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a

ellipse

b

circle

c

hyperbola

d

parabola

answer is D.

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Detailed Solution

$\begin{array}{l} M = (x, 0) N = (0, y - x y_{1}) \\ Δ O M N = 1 \Rightarrow x (y - x y_{1}) = 2 \\ \frac{d y}{d x} - \frac{y}{x} = \frac{- 2}{x^{2}} I. F = \frac{1}{x} \\ Sol is \frac{y}{x} = \int \frac{- 2}{x^{3}} d x = \frac{1}{x^{2}} + c \\ \Rightarrow x y - 1 = c x^{2} hyperbola \end{array}$

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