Question

I. lf the subnormal to the curve $x\times y^n=a^{n+1}$ is constant, then the value of $n$ is $-2$.
II. The length of the subtangent, ordinate of a point, length of the subnormal on a point other than the origin to the curve $y^2=4ax$ are in $G.P$.
Which of the above statements is correct?

• A. only I
• B. only II
• C. both I and II
• D. neither I nor II

Answer 1

The correct option is C both I and II

Statement I:

Subnormal to a curve at the point $(x_1,y_1)$ is given by $S_N=|y_{1}m|$ where $m$ is the slope of the curve at the point.

$x\times y^n=a^{(n+1)}$

$\therefore y=a^{\frac{n+1}{n}}\times x^{-\frac{1}{n}}$

$m={\frac{dy}{dx}|}_{(x_1,y_1)}=a^{\frac{n+1}{n}}\times (-\frac{1}{n})\times x_1^{-\left(\frac{1}{n+1}\right)}$

$S_N=|y_{1}m|=\frac{a^{\frac{2n+2}{n}}}{n}\times x_1^{-(\frac{2}{n}+1)}$

For $S_N$ to be constant, $\frac{2}{n}+1=0$

$\Rightarrow n=-2$

Statement 2:

The co-ordinates of a parabola $y^2=4ax$ in parametric form are $(a{t}^2,2at)$

Length of subnormal$=|y_{1}m|=2a{t}_1\times \frac{{\frac{dy}{dt}|}_{t_1}}{{\frac{dx}{dt}|}_{t_1}}=2a{t}_1\times \frac{1}{t_1}=2a$

Length of subtangent$=\left|\frac{y_1}{m}\right|=\frac{2a{t}_1}{1/t_1}=2a{t}_1^2$

Ordinate of the point$=2a{t}_1$

Therefore, the subtangent, ordinate and subnormal are in $G.P$ when $t_1\ne 0$

Hence, both statements are correct.