If A is the area of the ellipse 3x^2 + 4xy + 3y^2 = 1, area of the ellipse is 3*5^0.5A/pi. Then find the value of A.

Hi Siddharth,

The Area of an ellipse is Πab.

This is a very nice question, where if you note in the given ellipse equation:

If you replace "y" by "x" you'd get the same ellipse equation.

And hence the ellipse is summetric about the line y=x.

Also if you replace "x" by "-y" and "y" by "-x" you will again get the same ellipse equation.

And hence the ellipse is also symmetric about the line y=-x.

So, now we conclude that the given ellipse is centred at the origin, with it's axes as y=x, and y=-x.

Now we start solving the problem.

Solve the ellipse eqn with y=x, you will get two points of intersection viz (1/root[10],1/root[10]) and (-1/root[10,-1/root[10])

Next solve the ellipse eqn with y=-x. Points of intersection (1/root[2],-1/root[2]) and (-1/root[2],1/root[2])

So now distance between one of the pair is "a" and the distance between other pair is "b"

So a=2/root(5)

and b=2.

So now pi(ab) = given value. And hence you will get "A".

Best Regards,

Ashwin (IIT Madras).

I did not get the full question but I have found the area of the given ellipse and it is - pi/root(20)

differentiate the equation wrt x treating y as constant. Again differentiate the equation wrt y treating x as constant. You will get pair of lines whose intersection point will give you the center of ellipse. take a point (rcosk,rsink) on the ellipse. Put the point and then find max value of r by making a equation in r and applying maxima, this will give you length of major axis. Again apply minima and get the length of major axis. Apply the formula of area pi*a*b