If √(y+x) + √(y-x) = a, a > 1, d^2y/dx^2 is equal to : Option 1: -2/a Option 2: -a^2/2 Option 3: 2/a^2 Option 4: 2/a

Answer: 2/a^2. Solution: Squaring: 2y + 2√(y^2 - x^2) = a^2, so √(y^2 - x^2) = (a^2 - 2y)/2. Squaring again: 4y^2 - 4x^2 = a^4 - 4a^2 y + 4y^2, giving y = a^2/4 + x^2/a^2. dy/dx = 2x/a^2, so d^2y/dx^2 = 2/a^2.

Calculus

Continuity & Differentiability

Medium

If $\sqrt{y+x} + \sqrt{y-x} = a$ , $a > 1$ , $\frac{d^2y}{dx^2}$ is equal to :

$-\frac{2}{a}$

$-\frac{a^2}{2}$

$\frac{2}{a^2}$

$\frac{2}{a}$

✅ Correct Option: 3

Squaring: $2y + 2\sqrt{y^2 - x^2} = a^2$ , so $\sqrt{y^2 - x^2} = \frac{a^2 - 2y}{2}$ .

Squaring again: $4y^2 - 4x^2 = a^4 - 4a^2 y + 4y^2$ , giving $y = \frac{a^2}{4} + \frac{x^2}{a^2}$ .

$\frac{dy}{dx} = \frac{2x}{a^2}$ , so $\frac{d^2y}{dx^2} = \frac{2}{a^2}$ .

2026: 21 May Shift 2

If $y = \sqrt{12^x(3x^2+2)}$ , then $\dfrac{dy}{dx}$ will be

(consider $\log_e x = \log x$ )

2026: 29 May Shift 1

If $e^y(1+x)=1$ , then $\dfrac{d^2y}{dx^2}$ is:

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(Parametric equations)	$\left(\frac{dy}{dx}\right)$
(A) $x = \frac{2}{t}, y = 2t$	(I) $4t^2$
(B) $x = t^3, y = 3t + 2$	(II) $2(t+1)$
(C) $x = \log t, y = 2t^2$	(III) $-t^2$
(D) $x = e^t, y = 2te^t$	(IV) $t^{-2}$

Choose the correct answer from the options given below:

2026: 22 May Shift 2

If the function $f(x)=\begin{cases}\frac{e^{2\log_e x^2}-1}{x-1}, & x\ne 1\\ a, & x=1\end{cases}$ is continuous at x=1, then the value of a is:

2025: 26 May Shift 2

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$ , then the value of K is:

2025: 22 May Shift 2

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

List-I	List-II
(A) $\sin x$ is continuous on:	(I) $\mathbb{R} - \{0\}$
(B) $\tan x$ is continuous on:	(II) $\mathbb{R}$
(C) $\cot x$ is continuous on:	(III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$
(D) $x^{-n}, n \in \mathbb{N}$ is continuous on:	(IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

Choose the correct answer from the options given below:

If y+x+y−x=a\sqrt{y+x} + \sqrt{y-x} = ay+x​+y−x​=a, a>1a > 1a>1, d2ydx2\frac{d^2y}{dx^2}dx2d2y​ is equal to :

Solution

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Keyboard Shortcuts

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If $\sqrt{y+x} + \sqrt{y-x} = a$ , $a > 1$ , $\frac{d^2y}{dx^2}$ is equal to :