First slide

Equation of line in Straight lines

Question

The vertices of a $∆$ OBC are O (0, 0), B(–3, –1) and C(–1, –3). The equation of a line parallel to BC and intersecting sides OB and OC whose distance from the origin is $\frac{1}{2}$ , is

Moderate

A

$x + y + \frac{1}{\sqrt{2}} = 0$
B

$x + y - \frac{1}{\sqrt{2}} = 0$
C

$x + y - \frac{1}{2} = 0$
D

$x + y + \frac{1}{2} = 0$

Solution

The equation of line BC is x + y + 4 = 0
$∴$ Equation of a line parallel to BC is x + y + k = 0
This is at a distance $\frac{1}{2}$ from the origin.
$∴ |\frac{k}{\sqrt{2}}| = \frac{1}{2}$
$∴ k = \pm \frac{1}{\sqrt{2}}$
Since BC and the required line are on the same side of the origin
$∴ k = \frac{1}{\sqrt{2}}$
Hence, the required line is $x + y + \frac{1}{\sqrt{2}} = 0$

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