If the line
$(3 x + 14 y + 7) + k (5 x + 7 y + 6) = 0$
is passing through (0,0), then find the value of $k$ .

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Solution

Given line is $(3 x + 14 y + 7) + k (5 x + 7 y + 6) = 0$

$\Rightarrow (3 x + 14 y + 7) + (5 k x + 7 k y + 6 k) = 0$

$\Rightarrow 3 x + 5 k x + 14 y + 7 k y + 7 + 6 k = 0$

$\Rightarrow (3 + 5 k) x + (14 + 7 k) y + (7 + 6 k) = 0$

If it is passing through (0,0) then (0,0) should satisfy the equation of line as:

$\Rightarrow (3 + 5 k) \times 0 + (14 + 7 k) \times 0 + (7 + 6 k) = 0$

$\Rightarrow 7 + 6 k = 0$

$\Rightarrow k = - \frac{7}{6}$

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