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)

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