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The line, L, makes an angle of 30° with the positive direction of the x-axis - Scottish Highers Maths - Question 7 - 2022

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The line, L, makes an angle of 30° with the positive direction of the x-axis. Find the equation of the line perpendicular to L, passing through (0, -4),

Worked Solution & Example Answer:The line, L, makes an angle of 30° with the positive direction of the x-axis - Scottish Highers Maths - Question 7 - 2022

Step 1

Find the gradient of L

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Answer

The gradient (m) of the line L can be found using the tangent of the angle it makes with the x-axis. Therefore, we have:

m=tan⁡(30°)=13m = \tan(30°) = \frac{1}{\sqrt{3}}

Step 2

Find gradient of perpendicular line

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Answer

The gradient of the line that is perpendicular to L can be calculated using the property that the product of the gradients of two perpendicular lines equals -1. Let the gradient of the perpendicular line be denoted as m'. Thus:

m′=−1m=−3m' = -\frac{1}{m} = -\sqrt{3}

Step 3

Determine equation of line

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Answer

To find the equation of the line passing through the point (0, -4) with gradient -\sqrt{3}, we use the point-slope form of a line:

y−y1=m′(x−x1)y - y_1 = m'(x - x_1)

Substituting for (0, -4) and m':

y−(−4)=−3(x−0)y - (-4) = -\sqrt{3}(x - 0)

This simplifies to:

y+4=−3xy + 4 = -\sqrt{3}x

or, in standard form:

y=−3x−4y = -\sqrt{3}x - 4

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