Tangents PQ, PR are drawn from P = (5, 0) to the ellipse x^2/16 + y^2/9 = 1.The length of the chord OR is : (a)6/5 (b)8/5 (c)18/5 (d)4/5
Answers
Given : Tangents PQ, PR are drawn from P = (5, 0) to the ellipse x²/16 + y²/9 = 1.
To Find : The length of the chord QR is
(a)6/5 (b)8/5 (c)18/5 (d)4/5
Solution:
x²/16 + y²/9 = 1
Major axis is along x axis
and point P= ( 5 , 0) is also along x axis
Hence tangent will be symmetric about x axis
Let say point of tangency are ( h ,k) and ( h, -k)
Length of chord = 2k
x²/16 + y²/9 = 1
=> 2x/16 + (2y/9)dy/dx = 0
=> dy/dx = - 9x/16y
dy/dx = - 9x/16y
at h , k = -9h/16k
Slope = (k - 0) / (h - 5) = -9h/16k
=> 16k² = -9h² + 45h
=> 9h² + 16k² = 45h
x²/16 + y²/9 = 1
=> 9x² + 16y² = 144
( h , k) lies on this
=> 9h² + 16k² = 144
9h² + 16k² = 45h
9h² + 16k² = 144
=> 45h = 144
=> h = 144/45 = 16/5
9( 16/5)² + 16k² = 144
=> 144/25 + k² = 9
=> k² = 81/25
=> k = 9/5
2k = 2 * (9/5) = 18/5
length of the chord QR = 18/5
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