Question

radius of curvature of x2-y2=a2​

Answer 1

r=a

d=a/2

Step-by-step explanation:

x²-y³=a²

x²-y³=r²

r²=a²

r=a

r=2d

2d=a

d=a/2

Answer 2

We need to recall the following formula for the radius of curvature.

• $R=\frac{(1+(\frac{dy}{dx})^2 )^{3/2}}{|\frac{d^2y}{dx^2} |}$

Given:

$x^{2} -y^2=a^2$

$y^2=x^2-a^2$

$y=\sqrt{x^2-a^2}$

Differentiate w.r.t. $x$ , we get

$\frac{dy}{dx}=\frac{x}{\sqrt{x^2-a^2}}$

Again differentiate w.r.t. $x$ , we get

$\frac{d^2y}{dx^2}=-\frac{a^{2} }{(x^2-a^2)^{3/2}}$

From the formula of a radius of curvature, we get

$R=\frac{(1+(\frac{x}{\sqrt{x^2-a^2} })^2 )^{3/2}}{|-\frac{a^2}{(x^2-a^2)^{3/2}} |}$

$R=\frac{(1+(\frac{x}{\sqrt{x^2-a^2} })^2 )^{3/2}}{\frac{a^2}{(x^2-a^2)^{3/2}} }$

$R=\frac{(1+\frac{x^2}{x^2-a^2})^{3/2}}{\frac{a^2}{(x^2-a^2)^{3/2}} }$

$R=\frac{(\frac{x^2-a^2+x^2}{x^2-a^2})^{3/2}}{\frac{a^2}{(x^2-a^2)^{3/2}} }$

$R=\frac{\frac{(2x^2-a^2)^{3/2}}{(x^2-a^2)^{3/2}}}{\frac{a^2}{(x^2-a^2)^{3/2}} }$

$R={\frac{(2x^2-a^2)^{3/2}}{a^2}}$

Hence, the radius of curvature of $x^{2} -y^2=a^2$ is ${\frac{(2x^2-a^2)^{3/2}}{a^2}}$.