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Find the length of the chord of the ellipse X 2 /25+y 2 /16=1 whose middle point is (1/2,2/5) Please answer immediately with solution.

Find the length of the chord of the ellipse 
X2/25+y2/16=1 whose middle point is (1/2,2/5)
Please answer immediately with solution.

Grade:11

2 Answers

Arun
25763 Points
3 years ago
The diameter thro ( 1/2, 2/5 ) has equation y=(4/5)x 

The conjugate diameter has gradient m given by m*(4/5) = −16/25 → m=−4/5 

All chords parallel to this are bisected by y=(4/5)x so chord of interest has m=−4/5 

∴ its equation is y−2/5 = (−4/5)(x−1/2) → y = (4/5)(1−x) 

This intersects ellipse when x²/25 + (16/25)(1−x)²/16 = 1 

This simplifies to x²−x−12 = 0 which has solutions x=4 or −3 

Using chord equation, the ends of chord are thus (4,−12/5) & (−3,16/5) 

Length of chord is thus √( 7² + (28/5)² ) = (7/5)√41
Deepak Kumar Shringi
askIITians Faculty 4405 Points
3 years ago
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