Question

Find the point on the line $fraction numerator straight x plus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction$ at a distance $3 square root of 2$ from the point (1, 2, 3).

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Solution

Any point on the line

fraction numerator straight x plus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction space is space straight P left parenthesis 3 space straight r space minus space 2 comma space 2 straight r minus 1 comma space 2 straight r plus 3 right parenthesis.

Distance of P from the point Q(1, 2, 3) is

3 square root of 2

therefore

PQ space equals space 3 square root of 2 space space space space space space space space space space space space space space space rightwards double arrow space space space PQ squared space equals space 18

therefore

left parenthesis 3 straight r minus 2 minus 1 right parenthesis squared space plus space left parenthesis 2 straight r minus 1 minus 2 right parenthesis squared space space plus left parenthesis 2 straight r plus 3 minus 3 right parenthesis squared space equals space 18

therefore space space space space left parenthesis 3 straight r minus 3 right parenthesis squared plus left parenthesis 2 straight r minus 3 right parenthesis squared plus left parenthesis 2 straight r right parenthesis squared space equals space 18 therefore space space space space 9 straight r squared minus 18 straight r plus 9 plus 4 straight r squared minus 12 straight r plus 9 plus 4 straight r squared space equals space 18 therefore space space space space 17 straight r squared minus 30 straight r space equals space 0 space space space or space space space straight r left parenthesis 17 straight r minus 30 right parenthesis space equals space 0 therefore space space space space space space space space space space space space space space space space space space space space space straight r space equals space 0 comma space space 30 over 17.

when r = 0, P is (-2, -1, 3)
when

straight r space equals space 30 over 7 comma space space space straight P space is space open parentheses 90 over 17 comma space minus 2 comma space 60 over 17 minus 1 comma space 60 over 17 plus 3 close parentheses space space space space straight i. straight e. space space space open parentheses 56 over 17 comma space 43 over 17 comma space 111 over 17 close parentheses

therefore

required point is

left parenthesis negative 2 comma space minus 1 comma space 3 right parenthesis space or space open parentheses 56 over 17 comma space 43 over 17 comma space 111 over 17 close parentheses.

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