The sphere x^2+y^2+z^2−2x+6y+14z+3=0 meets the line joining A(2,−1,4),B(5,5,5) in the points C and D. Prove that AC/CB=−AD/DB=1/2.

Glenn Hopkins 2022-07-23 Answered
The sphere x 2 + y 2 + z 2 2 x + 6 y + 14 z + 3 = 0 meets the line joining A ( 2 , 1 , 4 ) , B ( 5 , 5 , 5 ) in the points C and D. Prove that A C : C B = A D : D B = 1 : 2
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Answers (2)

Sandra Randall
Answered 2022-07-24 Author has 17 answers
Hint : The equation of the line through A and B is ( x , y , z ) = ( 2 + 3 t , 1 + 6 t , 4 + t ). Substitute this into the equation for the circle, solve the quadratic to find the points C and D ... etc ...
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Livia Cardenas
Answered 2022-07-25 Author has 5 answers
The line through A ( 2 , 1 , 4 ) , B ( 5 , 5 , 5 ) has a direction given by u = B A = { 3 , 6 , 1 }
so the line A B has equation A + s v = { 2 + 3 s , 1 + 6 s , 4 + s }
plug in the equation of the sphere
( s + 4 ) 2 + ( 3 s + 2 ) 2 + ( 6 s 1 ) 2 2 ( 3 s + 2 ) + 6 ( 6 s 1 ) + 14 ( s + 4 ) + 3 = 0
2 ( 23 s 2 + 26 s + 35 ) = 0
Which has no real solutions
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