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.

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Grade 11Analytical Geometry

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.

supraja venkatraman

7 Years agoGrade 11

2 Answers

Arun

Approved Tutor Answer7 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