Question

Find the equation of the tangent to the curve at any point $(X,Y)$.

$\frac{x^m}{a^m}+\frac{y^m}{b^m}=1.$

• A. $\frac{X}{a}{\left(\frac{x}{a}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}=1$
• B. $\frac{X}{b}{\left(\frac{x}{a}\right)}^{m-1}+\frac{Y}{a}{\left(\frac{y}{b}\right)}^{m-1}=1$
• C. $\frac{X}{a}{\left(\frac{x}{b}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{a}\right)}^{m-1}=1$
• D. $\frac{X}{b}{\left(\frac{x}{b}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}=1$

Answer 1

The correct option is A $\frac{X}{a}{\left(\frac{x}{a}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}=1$

Given, $\frac{x^m}{a^m}+\frac{y^m}{b^m}=1.$ ..(1)
Differentiating w.r.t. x, we get
$m.\frac{x^{m-1}}{a^m}+m.\frac{y^{m-1}}{b^m}\cdot \frac{dy}{dx}=0.$
$\therefore \frac{dy}{dx}=\frac{1}{a}{\left(\frac{x}{a}\right)}^{m-1}.b{\left(\frac{b}{y}\right)}^{m-1}$
$\therefore$ Equation of tangent is $Y-y=\frac{dy}{dx}(X-x)$
or $Y-y=-\frac{1}{a}{\left(\frac{x}{a}\right)}^{m-1}.b{\left(\frac{b}{y}\right)}^{m-1}(X-x)$
or $\frac{X}{a}{\left(\frac{x}{a}\right)}^{m-1}-{\left(\frac{x}{a}\right)}^m=-\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}+{\left(\frac{y}{b}\right)}^m$
or $\frac{X}{a}{\left(\frac{x}{a}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}={\left(\frac{x}{a}\right)}^m+{\left(\frac{y}{b}\right)}^m$
or $\frac{X}{a}{\left(\frac{x}{a}\right)}^{m-1}+\frac{Y}{b}{\left(\frac{y}{b}\right)}^{m-1}=1,$ by (1)