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Question
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
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Solution
Let the sphere `x^2+y^2+z^2=504` meet the line joining the given points at the point (x1, y1, z1).
Then, we have:
`x_1^2+y_1^2+z_1^2=504` .............(1)
Suppose the point (x1, y1, z1) divides the line joining the points (12, – 4, 8) and (27, – 9, 18) in the ratio λ:1.
∴`x_1=(27 λ+12)/( λ+1), y_1=(-9λ-4)/(λ+1),z_1=(18λ+8)/(λ+1)`
Substitute these values in equation (1):
`(27λ+12)^2/(λ+1)^2+(-9λ-4)^2/(λ+1)^2+(18λ+8)^2/(λ+1)^2=504`
⇒` 9(9λ+4)^2+(9λ+4)^2+4(9λ+4)^2=504(λ+1)^2`
⇒`9λ+4=+-6(λ+1)`
⇒`λ=+-2/3`
Thus, the sphere divides the line joining the given points in the ratio 2:3 and 2:– 3.
Hence, the given sphere `x^2+y^2+z^2=504` divides the line joining the points (12, – 4, 8) and (27, – 9, 18) internally in the ratio 2:3 and externally in the ratio −2:3.
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