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

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

Question

If an ellipse slides on the rectangular coordinate axes and foci of the ellipse are at

(2,y_1)

and

(3,y_2)

, then

y_1y_2

equals

5

1

6

correct

\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

b^2

. Step 2: As the ellipse slides on the axes, both coordinate axes are tangents. For the

x

-axis the perpendicular distances of the foci are

y_1

and

y_2

; for the

y

-axis they are

x_1

and

x_2

. Hence

x_1x_2=y_1y_2=b^2.

Step 3: With

x_1=2

and

x_2=3

y_1y_2=x_1x_2=2\cdot3=6.

Correct answer: (3)

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