Mathematics

Given a straight line x cos 30° + y sin 30° = 2. Determine the equation of the other line which is parallel to it and passes through (4, 3).

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Given line equation,

⇒ x cos 30° + y sin 30° = 2

⇒ $\dfrac{\sqrt{3}}{2}x + \dfrac{y}{2} = 2$

⇒ y + $\sqrt{3}x$ = 4

⇒ y = - $\sqrt{3}x + 4$ .

Comparing above equation with y = mx + c we get,

Slope (m₁) = - $\sqrt{3}$ .

Since, slope of parallel lines are equal,

Slope of a line which is parallel to this given line = - $\sqrt{3}$ .

By slope-point form, the equation of the required line is,

⇒ y - y₁ = m(x - x₁)

⇒ y - 3 = - $\sqrt{3}$ (x - 4)

⇒ y - 3 = $-\sqrt{3}x + 4\sqrt{3}$

⇒ $\sqrt{3}x + y = 4\sqrt{3} + 3$ .

Hence, the equation of the required line is $\sqrt{3}x + y = 4\sqrt{3} + 3$ .

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