The angle between the curves y^2=x and x^2=y at (1,1) is: - Vidyadip

MCQ

The angle between the curves $y^2=x$ and $x^2=y$ at $(1,1)$ is:

A
$\tan^{-1}\frac{4}{3}$
✓
$\tan^{-1}\frac{3}{4}$
C
$90^\circ$
D
$45^\circ$

✓

Answer

Correct option: B.

$\tan^{-1}\frac{3}{4}$

Given:
$\text{y}^2=\text{x} \ ...(1)$
$\text{x}^2=\text{y} \ ...(2)$
Point $= (1, 1)$
On diffierentiating $(1) \text{w.r.t. x},$ we get
$2\text{y}\frac{\text{dy}}{\text{dx}}=1$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{2\text{y}}$
$\Rightarrow\text{m}_1=\frac{1}{2}$
On differentiating $(2) \text{w.r.t. x},$ we get
$2\text{x}=\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\text{m}_2(1)=2$
Now,
$\tan\theta=\Big|\frac{\text{m}_1-\text{m}_2}{1+\text{m}_1\text{m}_2}\Big|=\Big|\frac{\frac{1}{2}-2}{1+\frac{1}{2}\times2}\Big|=\frac{3}{4}$
$\Rightarrow\theta=\tan^{-1}\big(\frac{3}{4}\big)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Application of Derivatives→Application of Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Two vectors each of magnitudes $1$ unit are inclined at $60^{\circ}$ to each other. The difference of the vectors has a magnitude $...........?$

The value of the integral ${\int_{ - 1/2}^{1/2} {\left[ {{{\left( {\frac{{x + 1}}{{x - 1}}} \right)}^2} + {{\left( {\frac{{x - 1}}{{x + 1}}} \right)}^2} - 2} \right]} ^{1/2}}dx$ is

A box contains $6$ nails and $10$ nuts. Half of the nails and half of the nuts are rusted. If one iten is chosen ar random, the probability that it is rusted or is nail is

The function $\text{f(x)}=\sin^{-1}(\cos\text{x})$ is:

If $\text{y}=\frac{\text{ax}+\text{b}}{\text{x}^2+\text{c}},$ then $(2\text{xy}_1+\text{y})\text{y}_3=$

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\text{lRm}$ if l is perpendicular to $m$ for all $l, m \in L.$ Then$, R$ is:

The values of $A$ and $B$ such that the function $f(x) = \left\{ {\begin{array}{*{20}{c}}{ - 2\sin x,}&{x \le - \frac{\pi }{2}}\\{A\sin x + B,}&{ - \frac{\pi }{2} < x < \frac{\pi }{2}}\\{\cos x,}&{x \ge \frac{\pi }{2}}\end{array}} \right.$, is continuous everywhere are

Let $n$ be a natural number and let $a$ be a real number. The number of zeroes of $x^{2 n+1}-(2 n+1) x+a=0$ in the interval $[-1,1]$ is

The solution of the differential equation ${(x + y)^2}\frac{{dy}}{{dx}} = {a^2}$ is

If the lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular to each other then $k =$ ?