Let a tangent to the curve intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is __________
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Answer: 7
Solution
The curve is the ellipse x²/16 + y²/9 = 144/144, i.e., a = 4, b = 3.
A tangent to the ellipse x²/16 + y²/9 = 1 at point (4cosθ, 3sinθ) is:
(x·cosθ)/4 + (y·sinθ)/3 = 1
This meets x-axis at A = (4/cosθ, 0) and y-axis at B = (0, 3/sinθ).
Length AB² = (4/cosθ)² + (3/sinθ)² = 16sec²θ + 9csc²θ
= 16(1 + tan²θ) + 9(1 + cot²θ) = 25 + 16tan²θ + 9cot²θ
By AM-GM: 16tan²θ + 9cot²θ ≥ 2√(16·9) = 2·12 = 24
So AB² ≥ 25 + 24 = 49, giving AB ≥ 7.
The minimum length of AB is 7.
A tangent to the ellipse x²/16 + y²/9 = 1 at point (4cosθ, 3sinθ) is:
(x·cosθ)/4 + (y·sinθ)/3 = 1
This meets x-axis at A = (4/cosθ, 0) and y-axis at B = (0, 3/sinθ).
Length AB² = (4/cosθ)² + (3/sinθ)² = 16sec²θ + 9csc²θ
= 16(1 + tan²θ) + 9(1 + cot²θ) = 25 + 16tan²θ + 9cot²θ
By AM-GM: 16tan²θ + 9cot²θ ≥ 2√(16·9) = 2·12 = 24
So AB² ≥ 25 + 24 = 49, giving AB ≥ 7.
The minimum length of AB is 7.
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]