The line, L1, makes an angle of 30° with the positive direction of the x-axis - Scottish Highers Maths - Question 7 - 2022
Question 7
The line, L1, makes an angle of 30° with the positive direction of the x-axis.
Find the equation of the line perpendicular to L1, passing through (0, -4),
Worked Solution & Example Answer:The line, L1, 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 L1
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Answer
The gradient of the line L1, represented as m1, can be obtained from the angle it makes with the x-axis. Using the tangent function, we have:
m1 = tan(30°) = \frac{1}{\sqrt{3}}.
Step 2
Determine the gradient of the perpendicular line
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Answer
For two lines to be perpendicular, the product of their gradients must equal -1. Therefore, if m1 = \frac{1}{\sqrt{3}}, then the gradient of the perpendicular line, m2, is:
m2 = -\frac{1}{m1} = -\sqrt{3}.
Step 3
Find the equation of the line
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Answer
To find the equation of the line with gradient m2 that passes through the point (0, -4), we can use the point-slope form of a line:
y - y_1 = m(x - x_1).
Substituting the values, we get:
y - (-4) = -\sqrt{3}(x - 0),
which simplifies to:
y + 4 = -\sqrt{3}x.
Thus, the equation of the line can be re-arranged as:
y = -\sqrt{3}x - 4.
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