The sides of the rectangle of greatest area t | vedclass

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

Question

(B) The equation of the ellipse is $x^2+4y^2=64$,which can be written as $\frac{x^2}{64} + \frac{y^2}{16} = 1$.This is in the form $\frac{x^2}{a^2} +

Vedclass · Accepted Answer

$8\sqrt{2}, 4\sqrt{2}$

Vedclass · Answer

(B) The equation of the ellipse is $x^2+4y^2=64$,which can be written as $\frac{x^2}{64} + \frac{y^2}{16} = 1$.This is in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,where $a^2=64$ $(a=8)$ and $b^2=16$ $(b=4)$.Let a vertex of the rectangle in the first quadrant be $(x, y) = (a \cos 	heta, b \sin 	heta) = (8 \cos 	heta, 4 \sin 	heta)$.The sides of the rectangle are $2x$ and $2y$,so the area $A = (2x)(2y) = 4xy = 4(8 \cos 	heta)(4 \sin 	heta) = 128 \sin 	heta \cos 	heta = 64 \sin(2	heta)$.The area is maximum when $\sin(2	heta) = 1$,i.e.,$2	heta = 90^\circ$ or $	heta = 45^\circ$.Substituting $	heta = 45^\circ$,we get $x = 8 \cos 45^\circ = 8 	imes \frac{1}{\sqrt{2}} = 4\sqrt{2}$ and $y = 4 \sin 45^\circ = 4 	imes \frac{1}{\sqrt{2}} = 2\sqrt{2}$.The sides of the rectangle are $2x = 8\sqrt{2}$ and $2y = 4\sqrt{2}$.

The sides of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$ are:

Similar Questions

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x$-axis at $Q$. If $M$ is the midpoint of the line segment $PQ$,then the locus of $M$ intersects the latus rectums of the given ellipse at the points

Find the equation of the ellipse with its center at the origin,passing through the points $(-3, 1)$ and $(2, -2)$,given that $a > b$.

For the ellipse $4x^2 + y^2 - 8x + 2y + 1 = 0$,the focus and the equation of the directrix are respectively

Find the area of the quadrilateral formed by the tangents at the endpoints of the latus rectum of the ellipse $x^2 + 2y^2 = 2$.

The eccentricity of the ellipse $4x^2 + 9y^2 + 8x + 36y + 4 = 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module