Question

Which of the following statements is false?

• A. The length of sub-tangent to the curve $x^2y^2 = 16a^4$ at the point $(-2a, 2a)$ is $2a$
• B. $x + y = 3$ is a normal to the curve $x^2 = 4y$
• C. Curves $y = -4x^2$ and $y = e^{-x/2}$ are orthogonal
• D. If a $\in (-1, 0),$ then tangent at each point of the curve $y=\frac{2}{3}x^{3}-2ax^{2}+2x+5$ makes an acute angle with the positive direction of x-axis

Answer 1

Answer: (C)

$\left(a\right) \,x^{2}y^{2}=16a^{4}$
$L_{ST}=\left|\frac{y}{x_{T}}\right| \Rightarrow xy=4a^{2}$
$y+xy'=0$
$y'=\frac{-y}{x} ; L_{ST}=\left|\frac{y}{y / x}\right|=x \Rightarrow L_{ST}=2a$
$\therefore \left(a\right) $ is true.
$\left(b\right) \, \frac{dy}{dx}=\frac{x}{2}=1 i.e. x=2$
$\therefore \left(2, 1\right)$ is the point on the curve $x^{2} = 4y$ at which the normal is
$y-1=-1\left(x-2\right) i.e. x+y=3 \,\therefore \left(b\right)$ is true
$\left(c\right) \,y=-4x^{2}, y=e^{-x/2}$
The curves are non-intersecting
$\therefore $ curves are not orthogonal i.e. $\left(c\right)$ is false.
$\left(d\right) \,y=\frac{2}{3}x^{3}-2ax^{2}+2x+5$
$\frac{dy}{dx}=2x^{2}-4ax+2\left(x^{2}-2ax+1\right)$
$=2\left(x-a\right)^{2}+2-2a^{2} > 0$
$\left[\because a\in\left(-1, 0\right) \Rightarrow 0 < a^{2} < 1\right]$
$\therefore \left(d\right)$ is true