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Equation of the sphere whose center is on the Y axes and which passes through the points (1, - 2, 5) and (5, 1, - 4)

Equation of the sphere whose center is on the Y axes and which passes through the points (1, - 2, 5) and (5, 1, - 4) Correct Answer x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> - 4y - 38 = 0

Let the equation of the sphere be

x2 + y2 + z2 + 2gx + 2fy + 2hz + d = 0

Since the center is on the Y axis, we have (0, 1, 0)i.e g = h = 0, y = 1

Hence equation of the sphere is

x2 + y2 + z2 + 2fy + d = 0

Since it passes through points (1, - 2, 5), (5, 1, - 4) we have

1 + 4 + 25 - 4f + d = 0

And 25 + 1 + 16 + 2f + d = 0

i.e 4f - d = 30

And 2f + d = - 42

Which gives f = - 2

Hence, d = - 38

So, equation of the sphere is

x2 + y2 + z2 - 4y - 38 = 0

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