Question

radius of curvature
${x}^{2} - {y}^{2} = {a}^{2}$​

Answer 1

Topic :-

Radius of Curvature

Given :-

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

To Find :-

Radius of Curvature of given equation.

Formula to be Used :-

The formula for the radius of curvature at any point x for the curve y = f(x) is given by:

$\mathrm{R=\left | \dfrac{(1+(y')^2)^{3/2}}{y''} \right | }$

where,

$\mathrm{y'=\dfrac{dy}{dx} \:\: and}$

$\mathrm{y''=\dfrac{d^2y}{dx^2}}$

Solution :-

We will calculate value of y' and y'' for given function and then put the values in the formula.

Calculating y' and y'',

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

Differentiate it,

$\mathrm {\dfrac{d(x^2)}{dx}-\dfrac{d(y^2)}{dx}=\dfrac{d(a^2)}{dx}}$

$\mathrm {2x-2y\dfrac{dy}{dx}=0}$

$\mathrm {2x-2yy'=0}$

$\mathrm {yy'=x}$

$\mathrm {y'=\dfrac{x}{y}}$

Again differentiate it,

$\mathrm {y''=\dfrac{1}{y}}$

Now, put values in formula,

$\mathrm{R=\left | \dfrac{(1+(x/y)^2)^{3/2}}{1/y} \right | }$

$\mathrm{R=\left | y(1+(x/y)^2)^{3/2} \right | }$

Answer :-

So, Radius of Curvature of given curve is

$\mathrm{R=\left | y(1+(x/y)^2)^{3/2} \right | }$