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Condition for Two Lines to Be Parallel

If a line is ...

Question

If a line is parallel to $3 x + 4 y - 12 = 0$ is at a distance of $2$ units from a fixed point $(6, 0)$ , then the equation of line is/are

3 x + 4 y - 8 = 0

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3 x + 4 y - 18 = 0

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3 x + 4 y - 28 = 0

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3 x + 4 y - 38 = 0

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Solution

The correct option is C $3 x + 4 y - 28 = 0$
Required line equation is
$3 x + 4 y + k = 0$

Distance from $(6, 0)$ is $2$ units
$∴ \frac{| 3 (6) + 4 (0) + k |}{5} = 2$
$\Rightarrow 18 + k = \pm 10 \Rightarrow k = - 8 or - 28$

$∴$ Required line equations are
$3 x + 4 y - 8 = 0$ and $3 x + 4 y - 28 = 0$

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