Question

If A, B, C, D are consecutive vertices of a rectangle whose area is 2006. An ellipse with area 2006π passes through A and C and has foci at B and D. then the eccentricity of the ellipse is​

Answer 1

Answer:

Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)

Step-by-step explanation:

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Answer 2

Given:

Consecutive vertices = A , B , C and D

Area of rectangle = 2006

Area of ellipse = 2006π

To Find:

Eccentricity of the ellipse

Solution:

Let the length of rectangle be = l and w

Let the axis of the ellipse on which the foci lie have length =2a

Let the length of other axis be = 2b

Now,

lw = ab = 2006

As per ellipse = l + w = 2a = l + w/2 = a

Diagnol of rectangle = √ l² + w²

a² = l² + w²/4 + b²

= l² + 2lw + w²/4  

= l² + w²/4 + b

= lw/2 = b²

b = √1003

Since ab = 2006 and a = √1003

Therefore,

a = 2√1003

Now, one fourth of the perimeter, thus -

1/4 × 2√1003

= 8√1003

Answer: The eccentricity of ellipse is 8√1003