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Sliding Ellipse on Axes: Product of Focal y-coordinates | JEE

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Question
If an ellipse slides on the rectangular coordinate axes and foci of the ellipse are at (2,y1)(2,y_1) and (3,y2)(3,y_2), then y1y2y_1y_2 equals
A55
B11
C66correct
D32\dfrac{3}{2}
Solution
Step 1: Use the focal property. The product of the perpendicular distances from the two foci onto any tangent of an ellipse is the constant b2b^2. Step 2: As the ellipse slides on the axes, both coordinate axes are tangents. For the xx-axis the perpendicular distances of the foci are y1y_1 and y2y_2; for the yy-axis they are x1x_1 and x2x_2. Hence
x1x2=y1y2=b2.x_1x_2=y_1y_2=b^2.
 Step 3: With x1=2x_1=2 and x2=3x_2=3,
y1y2=x1x2=2⋅3=6.y_1y_2=x_1x_2=2\cdot3=6.
 Correct answer: (3)
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