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Major Axis of Ellipse Tangent to x-axis, Foci Given | JEE

JEE Maths question with a full step-by-step solution.

Question
An ellipse has foci at (9,20)(9,20) and (49,55)(49,55) in the xyxy-plane and is tangent to the xx-axis. The length of its major axis is:
A8585correct
B7575
C6565
D5555
Solution
Step 1: Use the reflection property. If an ellipse is tangent to a line, the reflection of one focus in that line, joined to the other focus, has length 2a2a. Step 2: Reflect F1=(9,20)F_1=(9,20) in the xx-axis to get F1′=(9,−20)F_1'=(9,-20). Step 3: Then
2a=F1′F2=(49−9)2+(55+20)2=402+752=1600+5625=7225=85.2a=F_1'F_2=\sqrt{(49-9)^2+(55+20)^2}=\sqrt{40^2+75^2}=\sqrt{1600+5625}=\sqrt{7225}=85.
 ∴\therefore the major axis has length 8585. Correct answer: (1)
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