Question

For the curve $4x^5 = 5y^4$, the ratio of the cube of the subtangent at a point on the curve to the square of the subnormal at the same point is

• A. $\left(\frac {4}{5}\right)^4$
• B. $\left(\frac {5}{4}\right)^4$
• C. $x\left(\frac {4}{5}\right)^4$
• D. $y\left(\frac {5}{4}\right)^4$

Answer 1

Answer: (A)

$\left(\frac {4}{5}\right)^4$

Answer 2

The given curve is $4 x^{5}=5 y^{4}$
$\Rightarrow 20 x^{4}=20 y^{3} \cdot \frac{d y}{d x}$
$\Rightarrow \frac{d y}{d x}=\frac{x^{4}}{y^{3}}$
We know that
Length of subnormal $(S N)=\left(y \cdot \frac{d x}{d y}\right)=\left(\frac{y^{4}}{x^{4}}\right)$
Length of subtangent $(S T)=\left(y \cdot \frac{d y}{d x}\right)=\left(\frac{x^{4}}{y^{2}}\right)$
But given condition is
$\frac{(S N)^{3}}{(S T)^{2}}=\frac{\left(y^{4} / x^{4}\right)^{3}}{\left(x^{4} / y^{2}\right)^{2}}=\left(\frac{y^{4}}{x^{4}}\right)^{3} \times\left(\frac{y^{2}}{x^{4}}\right)^{2}$
$=\frac{y^{12}}{x^{12}} \times \frac{y^{4}}{x^{8}}=\left(\frac{y^{16}}{x^{20}}\right)$
$=\left(\frac{y^{4}}{x^{5}}\right)^{4}$
$=\left(\frac{4}{5}\right)^{4}=\frac{4^{4}}{5^{4}}$
$\left(\because 4 x^{5}=5 y^{4}\right)$