Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points - Maths - Introduction to Three Dimensional Geometry

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) PLZZZZZZ GIVE THE ANSWER BY TODAY
ITSELF!!!!!!!!!!!!!!!! I will give thumbs up to those who will give
me the answer!!!!!!!!!!!!!!!!

Santosh Kumar · Accepted Answer

Let the sphere $x^{2} + y^{2} + z^{2} = 504$ meet the line joining the given points at the point (x₁, y₁, z₁).

Then,

$x_{1}^{2} + y_{1}^{2} + z_{1}^{2} = 504 ...... (1)$

Suppose the point (x₁, y₁, z₁) divides the join of the points (12, – 4, 8) and (27, – 9, 18) in the ratio λ : 1.

Then,

$x_{1} = \frac{27 λ + 12}{λ + 1}, y_{1} = \frac{- 9 λ - 4}{λ + 1}, z_{1} = \frac{18 λ + 8}{λ + 1}$

Putting these values in equation (1),

$\frac{{(27 λ + 12)}^{2}}{{(λ + 1)}^{2}} + \frac{{(- 9 λ - 4)}^{2}}{{(λ + 1)}^{2}} + \frac{{(18 λ + 8)}^{2}}{{(λ + 1)}^{2}} = 504 \Rightarrow 9 (9 λ + 4)^{2} + {(9 λ + 4)}^{2} + 4 {(9 λ + 4)}^{2} = 504 {(λ + 1)}^{2} \Rightarrow 9 λ + 4 = \pm 6 (λ + 1) \Rightarrow λ = \pm \frac{2}{3}$

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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Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). PLZZZZZZ GIVE THE ANSWER BY TODAY ITSELF!!!!!!!!!!!!!!!!. I will give thumbs up to those who will give me the answer!!!!!!!!!!!!!!!!.

Find the ratio in which the sphere x² + y² + z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). PLZZZZZZ GIVE THE ANSWER BY TODAY ITSELF!!!!!!!!!!!!!!!!. I will give thumbs up to those who will give me the answer!!!!!!!!!!!!!!!!.