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Combination Based on Geometry

At any point ...

Question

At any point on the curve y $=$ f(x), the length of the sub-normal is constant, then the curve is

circle

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ellipse

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parabola

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straight line

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Solution

The correct option is C parabola
SN $=$ (ym)
y m $=$ k
$y f^{'} (x) = k$
$\frac{y d y}{d x} = k$
$y d y = k d x$
$\frac{y^{2}}{2} = k x + c$
$y^{2} = 2 k x + 2 c$
This a parabola.

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