Question

Let curve $C$ is such that length of subnormal at any point of the curve is 2 and curve passes through the point $P(1,2)$, and $Q(4,4)$ then

• A. length of subtangent at point $P$ is 2
• B. area bounded by $C$ with line $x=1$ is $\frac{8}{3}$
• C. length of subtangent at point $P$ is 4
• D. area bounded by $C$ with line $x=1$ is $\frac{4}{3}$

Answer 1

Answer: (A), (B)

$|my|=2\Rightarrow ydy=2dx\Rightarrow \frac{y^2}{2}=2x+C$
$\therefore y^2=4x\Rightarrow P\left(t^2,2t\right)$
$\therefore ST=\frac{\frac{2t}{2}}{\frac{2}{2t}}=2t^2=2$

$A=2\underset{0}{\overset{1}{\int }}2\sqrt{x}dx=\frac{4\cdot {\left(x^{\frac{3}{2}}\right)}_0^1}{\frac{3}{2}}=\frac{8}{3}$
$\Theta y\frac{dy}{dx}=2$
$\therefore$ Orthogonal trajectory is $y\frac{dx}{dy}=-2\Rightarrow -2\frac{dy}{y}=dx\Rightarrow -2\ln y=x+\lambda$
$\Rightarrow y=\alpha \cdot e^{\frac{-x}{2}}$.