(1) A circle is concentric with the ellipse  (x^2)/(a^2) + (y^2)/(b^2) =1 and passed through the focus F1 and F2 of the ellipse.  Two curves intersect at four points. Let  P  be any point of intersection. If the major axis of the ellipse is  15  and the area of the triangle PF1F2  = 26 , then find the value of 4a^2  - 4b^2 .

Asked by sudhanshubhushanroy | 20th Mar, 2018, 10:12: PM

Expert Answer:

begin mathsize 16px style Given space ellipse space is space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space circle space straight x squared plus straight y squared equals straight a squared minus straight b squared
Major space axis space is space 2 straight a equals 15 rightwards double arrow straight a equals 7.5
straight x squared plus straight y squared equals straight a squared minus straight b squared rightwards double arrow straight x squared equals straight a squared minus straight b squared minus straight y squared
Put space it space in space straight x squared over straight a squared plus straight y squared over straight b squared equals 1
fraction numerator straight a squared minus straight b squared minus straight y squared over denominator straight a squared end fraction plus straight y squared over straight b squared equals 1 space find space straight y equals fraction numerator straight b squared over denominator square root of straight a squared minus straight b squared end root end fraction
Area space of space triangle equals 26
1 half cross times 2 square root of straight a squared minus straight b squared end root cross times fraction numerator straight b squared over denominator square root of straight a squared minus straight b squared end root end fraction equals 26
rightwards double arrow straight b equals square root of 26
rightwards double arrow 2 square root of straight a squared minus straight b squared end root equals 2 square root of open parentheses 15 over 2 close parentheses squared minus 26 end root
rightwards double arrow 2 square root of straight a squared minus straight b squared end root equals 11
rightwards double arrow 4 open parentheses straight a squared minus straight b squared close parentheses equals 121 end style

Answered by Sneha shidid | 21st Mar, 2018, 01:14: PM