Rectangle in an Ellipse

Geometry Level 2

$A,B,C,D$ are consecutive vertices of a rectangle whose area is $2006$ square units. An ellipse with area $2006\pi$ , which passes through $A$ and $C$ has its foci at $B$ and $D$ .

If the perimeter of the rectangle can be expressed as $p\sqrt{q}$ where $p$ is a positive integer and $q$ is a square-free positive integer, find $p+q$ .

The answer is 1011.

1 solution

Trevor Arashiro
Apr 26, 2015

Let a and b be the major and minor axes respectively.

$ab=2006$

Let x,y be the sides of the rectangle

$xy=2006$

Since it's a rectangle, its got a right angle. Thus the distance between the foci is

$2\sqrt{a^2-b^2}$

Using Pythagorean theorem. We have

$x^2+y^2=4(a^2-b^2)$

By The definition of an ellipse, we have $x+y=2b$ since the total distance from both foci is positive.

Solving these equations gives $b=2\sqrt{1003}$

Thus since b is 1/4 the perimeter, the perimeter is $8\sqrt{1003}$ so $8+1003=1011$

There is a small typo it should be $x+y=2a$ which yields $b=\sqrt{1003}$

Seong Ro - 5 years, 5 months ago

why did u took a & b as major as semi major

Karan Kanojia - 5 years, 1 month ago

I just used two arbitrary variables. They are not the ones from $a\sqrt{b}$

Trevor Arashiro - 5 years, 1 month ago

Rectangle in an Ellipse

The answer is 1011.

1 solution

0 pending reports