The two curves x=y^2 and xy=a^3 cut orthogona | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given curves are $x=y^2$ $(i)$ and $xy=a^3$ (ii).For curve $(i)$,differentiating with respect to $x$: $1 = 2y \frac{dy}{dx} \Rightarrow \frac{dy}{

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(B) Given curves are $x=y^2$ $(i)$ and $xy=a^3$ (ii).For curve $(i)$,differentiating with respect to $x$: $1 = 2y \frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{1}{2y}$.For curve (ii),differentiating with respect to $x$: $x \frac{dy}{dx} + y = 0 \Rightarrow \frac{dy}{dx} = -\frac{y}{x}$.To find the point of intersection,substitute $x=y^2$ into $xy=a^3$: $y^2(y) = a^3 \Rightarrow y^3 = a^3 \Rightarrow y = a$.Then $x = a^2$. So the point of intersection is $(a^2, a)$.The slopes at $(a^2, a)$ are $m_1 = \frac{1}{2a}$ and $m_2 = -\frac{a}{a^2} = -\frac{1}{a}$.Since the curves cut orthogonally,$m_1 m_2 = -1$.Therefore,$(\frac{1}{2a})(-\frac{1}{a}) = -1 \Rightarrow -\frac{1}{2a^2} = -1 \Rightarrow 2a^2 = 1 \Rightarrow a^2 = \frac{1}{2}$.

The two curves $x=y^2$ and $xy=a^3$ cut orthogonally at a point,then $a^2$ is equal to

Similar Questions

The correct order of increasing $s-$character (in percentage) in the hybrid orbitals of the following molecules/ions is:
$(I)$ $CO_3^{2-}$ $(II)$ $XeF_4$ $(III)$ $I_3^-$ $(IV)$ $NCl_3$ $(V)$ $BeCl_2$

Which of the following groups of cells in the male gonad represents haploid cells?

$A$ bryophyte,which harbours a nitrogen-fixing blue-green alga in its thallus,is

An unbiased coin is tossed to get $2$ points for turning up a head and $1$ point for the tail. If three unbiased coins are tossed simultaneously,then the probability of getting a total of an odd number of points is

If $\overrightarrow{u}=\overrightarrow{a}-\overrightarrow{b}$,$\overrightarrow{v}=\overrightarrow{a}+\overrightarrow{b}$,$|\overrightarrow{a}|=|\overrightarrow{b}|=2$,then $|\overrightarrow{u} \times \overrightarrow{v}|$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module