Question

$A:$ Length of sub-normal to $y=4x^2$ at $(1,4)$.
$B:$ Length of sub-normal to $y=x^3+1$ at $(1,2)$.
$C:$ Length of sub-tangent to $xy=12$ at $(3,4)$.
$D:$ Length of normal to $x^3+y^3=6xy$ at $(3,3)$.
The ascending order of the above values is

• A. $DCBA$
• B. $ABCD$
• C. $CDBA$
• D. $ABDC$

Answer 1

The correct option is C $CDBA$

(A) $y^{{}^{\prime }}=8x$
${y^{\prime }|}_{x=1}=8$

$SN=|ym|=4\times 8=32$

(B) $y^{\prime }=3x^2$
${y^{\prime }|}_{x=1}=3$
$SN=|ym|=3\times 2=6$

(C) $y^{\prime }=\frac{-12}{x^2}$
${y^{\prime }|}_{x=3}=\frac{-4}{3}$
$ST=\left(\frac{y}{m}\right)=\left(\frac{4+3}{4}\right)=3$

(D) $x^2+y^2{y}^{\prime }=2(x{y}^{\prime }+y)$
$9+9y^{\prime }=2(3y^{\prime }+3)\Rightarrow 3+3y^{\prime }=2y^{\prime }+2\Rightarrow y^{{}^{\prime }}=-1$
$N=\left|y\sqrt{1+m^2}\right|=3\sqrt{2}$

Ascending order of values
$C<D<B<A$
Hence, option C.