A line passing through the point A(9,0) makes an angle of 30∘ with the positive direction of x-axis. If this line is rotated about A through an angle of 15∘ in the clockwise direction, then its equation in the new position is
Question
A line passing through the point A(9,0) makes an angle of 30∘ with the positive direction of x-axis. If this line is rotated about A through an angle of 15∘ in the clockwise direction, then its equation in the new position is
Solution
The equation of a line in a plane is given by y = mx + c, where m is the slope of the line and c is the y-intercept.
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First, we need to find the slope of the line that makes an angle of 30 degrees with the positive direction of the x-axis. The slope (m1) of this line is given by the tangent of the angle it makes with the x-axis. So, m1 = tan(30) = 1/√3.
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The line is then rotated about point A(9,0) through an angle of 15 degrees in the clockwise direction. The slope (m2) of the line in the new position is given by the tangent of the sum of the original angle and the angle of rotation. Since the rotation is in the clockwise direction, we subtract the angle of rotation from the original angle. So, m2 = tan(30 - 15) = tan(15).
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Now, we can find the equation of the line in the new position. Since the line passes through the point A(9,0), its y-intercept is 0. So, the equation of the line is y = m2x + 0.
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Substituting the value of m2, we get y = tan(15)x. This is the equation of the line in the new position.
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