Question

Find the curve for which the sum of the lengths of the tangent and subtangent at any of its point is proportional to the product of the coordinates of the point of tangency, the proportionality factor is equal to k.

• A. $y=\frac{1}{k}ln|c(k^2{x}^2-1)|$
• B. $y=\frac{1}{k}ln|c(k{x}^2-1)|$
• C. $y=\frac{1}{k}ln|c(k{x}^2+1)|$
• D. $y=\frac{1}{k}ln|c(k^2{x}^2+1)|$

Answer 1

The correct option is A $y=\frac{1}{k}ln|c(k^2{x}^2-1)|$

Length of tangent at any point of curve $=y\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}$
Length of subtangent $=y\frac{dx}{dy}$
$y\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}+y\frac{dx}{dy}=kxy$
$\Rightarrow \sqrt{1+{\left(\frac{dx}{dy}\right)}^2}=kx-\frac{dx}{dy}$
$\Rightarrow 1+{\left(\frac{dx}{dy}\right)}^2=k^2{x}^2+{\left(\frac{dx}{dy}\right)}^2-2kx\frac{dx}{dy}$
$\Rightarrow 2kx\frac{dx}{dy}=k^2{x}^2-1\Rightarrow \int \frac{2kxdx}{k^2{x}^2-1}=\int dy$
$\Rightarrow =\frac{1}{k}ln|c(k^2{x}^2-1)|$