The area of the region x,y:xy≤8, 1≤y≤x2 is

JEE Advanced · Mathematics · 28. Area Under Curves

The area of the region $\{(x, y) : x y \leq 8, 1 \leq y \leq x^{2}\}$ is

A $16 \log_{e} 2 - \frac{14}{3}$
B $8 \log_{e} 2 - \frac{7}{3}$
C $8 \log_{e} 2 - \frac{14}{3}$
D $16 \log_{e} 2 - 6$

Verified Solution

Answer & Solution

Correct Answer

(A) $16 \log_{e} 2 - \frac{14}{3}$

Step-by-step Solution

Detailed explanation

To draw the inequality, let us draw the equation

x y = 8

and

y = 1

and

y = x^{2}

For point of intersection
(i)

x y = 8

and

y = 1 \Rightarrow A (8,1)

(ii)

x y = 8

and

y = x^{2} \Rightarrow x^{3} = 8 \Rightarrow x = 2 \Rightarrow B (2, 4)

(iii)

y = x^{2}

and

y = 1 \Rightarrow C (1, 1)

and

D (1, 1)

Now

x y \leq 8 \Rightarrow

region which contains origin

y \geq 1 \Rightarrow

region above line

y = 1

y \leq x^{2} \Rightarrow

region outside the parabola
Now required area
Method I:
Using x-axis:

A = \int_{1}^{2} (x^{2} - 1) \cdot d x + \int_{2}^{8} (\frac{8}{x} - 1) \cdot d x

A = {[\frac{x^{3}}{3} - x]}_{1}^{2} + {[8 l n x - x]}_{2}^{8}

A = - \frac{14}{3} + 16 l n 2

Method II:
Using y-axis:

A = \int_{1}^{4} (\frac{8}{y} - \sqrt{y}) \cdot d y

A = {[8 l n y - \frac{2}{3} y^{\frac{3}{2}}]}_{1}^{4} \Rightarrow A = - \frac{14}{3} + 16 l n 2

Note: The question should include bounded area term as in

2^{n d}

quadrant there exist a area which satisfy the inequality and is unbounded.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Paragraph:
A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation
$\sqrt{3} x+y-6=0$ and the point $D$ is $\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.Question:
The equation of circle $C$ is

JEE Advanced 2008 Easy

Match the statements of Column I with these in Column II.
[Note : Here $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and the real part of $z$ ]

Column I	Column II
(A)The set of points $z$ satisfying $\|z-i\| z\|\|=\|z+i\| z\|\|$ is contained in or equal to	(p)an ellipse with eccentricity $4 / 5$
(B) The set of points $z$ satisfying $\|z+4\|+\|z-4\|=0$ is contained in or equal to	(q) the set of points $z$ satisfying $\operatorname{Im}(z)=0$
(C) If $\|w\|=2$, then the set of points $z=w-\frac{1}{w}$ is contained in or equal to	(r) the set of points $z$ satisfying $\|\operatorname{Im} z\| \leq 1$
(D) If $\|w\|=1$, then the set of points $z=w+\frac{1}{w}$ is contained in or equal to	(s) the set of points satisfying $\|\operatorname{Re} z\| \leq 2$
	(t) the set of points $z$ satisfying $\|z\| \leq 3$

JEE Advanced 2010 Hard

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Column I	Column II
(A)The set of points \(z\) satisfying \(\|z-i\| z\|\|=\|z+i\| z\|\|\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(\|z+4\|+\|z-4\|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(\|w\|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(\|\operatorname{Im} z\| \leq 1\)
(D) If \(\|w\|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(\|\operatorname{Re} z\| \leq 2\)
	(t) the set of points \(z\) satisfying \(\|z\| \leq 3\)

	LIST -I		LIST-II
A)	$\vec{r} (t) = α t \hat{i} + β \hat{j}$	P)	$\vec{p}$
B)	$\vec{r} (t) = α \cos ω t \hat{i} + β \sin ω t \hat{j}$	Q)	$\vec{L}$
C)	$\vec{r} (t) = α \cos ω t \hat{i} + \sin ω t \hat{j}$	R)	$K$
D)	$\vec{r} (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j}$	S)	$U$
		T)	$E$