Question

If the line x/3=y/4=z is perpendicular to the line (x-1)/k=(y+2)/3=(z-3)/(k-1) then find the value of 'k'

Answer 1

SOLUTION

TO DETERMINE

The value of k such that below two lines are perpendicular

$\displaystyle \sf{ \frac{x}{3} = \frac{y}{4} = z \: \: and \: \: \frac{x - 1}{k} = \frac{y + 2}{3} = \frac{z - 3}{k - 1} }$

EVALUATION

Here the given equation of the lines are

$\displaystyle \sf{ \frac{x}{3} = \frac{y}{4} = z \: \: and \: \: \frac{x - 1}{k} = \frac{y + 2}{3} = \frac{z - 3}{k - 1} }$

Which can be rewritten as

$\displaystyle \sf{ \frac{x}{3} = \frac{y}{4} = \frac{z - 0}{1} \: \: and \: \: \frac{x - 1}{k} = \frac{y + 2}{3} = \frac{z - 3}{k - 1} }$

Since the two lines are perpendicular

$\displaystyle \sf{ (3 \times k) + (4 \times 3) + (1 \times (k - 1)) = 0}$

$\displaystyle \sf{ \implies \: 3k + 12 + k - 1 = 0 \: }$

$\displaystyle \sf{ \implies \: 4k + 11 = 0 \: }$

$\displaystyle \sf{ \implies \: 4k = - 11 \: }$

$\displaystyle \sf{ \implies \: k = - \frac{11}{4} \: }$

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