The curve for which the normal at any point $P (x, y)$ and the line joining origin to the point $P$ form an isosceles triangle with $x -$ axis as base is

An ellipse

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A parabola

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A rectangular hyperbola

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A circle

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Solution

The correct option is D A circle
It is given that the triangle $O P G$ is an isosceles triangle.
Therefore, $O M = M G =$ sub-normal
$\Rightarrow x = ∣ ∣ ∣ y \frac{d y}{d x} ∣ ∣ ∣ \Rightarrow x d x = \pm y d y$
On intergating both sides, we get $x^{2} - y^{2} = C,$ or $x^{2} + y^{2} = c$ which is a rectangular hyperbola or a circle.

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The curve for which the normal at any point P(x, y) and the line joining origin to the point P form an isosceles triangle with x−axis as base is

The correct option is D A circleIt is given that the triangle OPG is an isosceles triangle. Therefore, OM=MG= sub-normal ⇒x=∣∣∣ydydx∣∣∣⇒xdx=±ydy On intergating both sides, we get x2−y2=C, or x2+y2=c which is a rectangular hyperbola or a circle.

The curve for which the normal at any point $P (x, y)$ and the line joining origin to the point $P$ form an isosceles triangle with $x -$ axis as base is