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)

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

$A=2 \int\limits_0^1 2 \sqrt{x} d x=\frac{4 \cdot\left(x^{\frac{3}{2}}\right)_0^1}{\frac{3}{2}}=\frac{8}{3}$
$\Theta y \frac{d y}{d x}=2$
$\therefore$ Orthogonal trajectory is $y \frac{ dx }{ dy }=-2 \Rightarrow-2 \frac{ dy }{ y }= dx \Rightarrow-2 \ln y = x +\lambda$
$\Rightarrow y =\alpha \cdot e ^{\frac{- x }{2}}$.