Question

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

• A. an ellipse
• B. a rectangular hyperbola
• C. a circle
• D. parabola

Answer 1

Answer: (B)

a rectangular hyperbola

Let the equation of normal be
$(y-y_0)=-\frac{dx}{dy}(x-x_0)$
$\frac{dy}{dx}=\frac{x_0-x}{y-y_0}$
The point at which it cuts $x$ axis in
$x=x_0+y_0\frac{dy}{dx}$
As it in isosceles ole.
$\Rightarrow \sqrt{x^2+y^2}=\sqrt{(y_0{y}_1{)}^2+y^2+(x_0-x{)}^2+2(x_0-x)y_0{y}^1}$
$\Rightarrow x^2=y_0{y}_1^2+(x_0-x{)}^2+2(x_0-x)y_0{y}^1$