P is a variable point which moves such that 3 P A = 2 P B. If A = \left(\right. - 2 , 2 , 3 \left.\right) and B = \left(\right. 13 , - 3 , 13 \left.\right) prove that P satisfies the equation x^{2} + y^{2} + z^{2} + 28 x - 12 y + 10 z

Engineering

Mathematics

Line of Greatest Slope

Question

P

is a variable point which moves such that

3 P A = 2 P B

. If

A = (- 2, 2, 3)

and

B = (13, - 3, 13)

prove that

P

satisfies the equation

x^{2} + y^{2} + z^{2} + 28 x - 12 y + 10 z - 247 = 0

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Let

P = (x, y, z) < / d i v > < d i v >

\text PA= \sqrt{(x-(-2))^2+(y-2)^2+(z-3)^2}\\

< / d i v > < d i v >

\text PB=\sqrt{(x-13)^2+(y-(-3))^2+(z-13)^2}\\

< / d i v > < d i v >

\text {given that}

3PA=2PB

< / d i v > < d i v > < b r / > < / d i v > < d i v >

3 \sqrt{(x+3)^2+(y-2)^2+(z-3)^2}=2\sqrt{(x-13)^2+(y+3)^2+(z-13)^2}\\

< / d i v > < d i v >

9[(x+3)^2+(y-2)^2+(z-3)^2]=4[(x-13)^2+(y+3)^2+(z-13)^2]\\

< / d i v > < d i v > o n s o l v i n g t h i s, w e g e t

x^2+y^2+z^2+28x-12y+10z-247=0$$

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