Question

The sum of the lengths of the tangent and normal at point $(2,2)$ on the curve $x^5=2y^4$ is

• A. $\frac{18\sqrt{41}}{10}$ units
• B. $\frac{9\sqrt{41}}{10}$ units
• C. $\frac{3\sqrt{41}}{10}$ units
• D. $\frac{\sqrt{41}}{10}$ units

Answer 1

The correct option is B $\frac{9\sqrt{41}}{10}$ units

$x^5=2y^4\Rightarrow 5x^4=8y^3\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{5x^4}{8y^3}$

$m={\left(\frac{dy}{dx}\right)}_{(2,2)}=\frac{5}{4}$

Required sum$=\left|2\sqrt{1+\frac{16}{25}}\right|+\left|2\sqrt{1+\frac{25}{16}}\right|$
$=\frac{9\sqrt{41}}{10}$ units.