If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to line, x/1 = y/4= z/5 is Q, then PQ is equal to

$6 square root of 7$
$3 square root of 5$
$2 square root of 42$
$square root of 42$

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Solution

2 square root of 42

Line space PQ space semicolon space fraction numerator straight x minus 1 over denominator 1 end fraction space equals fraction numerator straight y plus 2 over denominator 4 end fraction space equals space fraction numerator straight z minus 3 over denominator 5 end fraction Let space straight F space left parenthesis straight lambda space plus 1 comma space 4 straight lambda minus 2 comma space 5 straight lambda plus 3 right parenthesis straight F space lies space on space the space plane 2 left parenthesis straight lambda plus 1 right parenthesis space plus space 3 left parenthesis 4 straight lambda minus 2 right parenthesis minus 4 left parenthesis 5 straight lambda plus 3 right parenthesis space plus space 22 space equals space 0 rightwards double arrow space minus 6 straight lambda space plus space 6 space equals space 0 rightwards double arrow space straight lambda space equals 1 straight F space left parenthesis 2 comma 2 comma 8 right parenthesis PQ space equals space 2 PF space equals space 2 square root of 42

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