Find the Equation of the Line in the New Position :
Here we are going to see how to find the equation of the line in the new position if a line joining two points is rotated about the point in counter clockwise direction through given angle.
Question 1 :
If the line joining two points A(2,0) and B(3,1) is rotated about A in anticlockwise direction through an angle of 15°, then find the equation of the line in new position.
Let us represent the given information in a picture.
Let A (2, 0) and B (3, 1) be the given points.
Slope of PQ = (y2 − y1) / (x2 − x1)
= (1 - 0) / (3 - 2)
= 1 ⇒ the angle of inclination of
the line AB = tan−1(1) = π/4 = 45◦
The slope of the line in new position is
m = tan(45◦ + 15◦)
Slope = tan(60◦) = (√3)
Equation of the straight line passing through (2, 0) and with the slope √3 is
y − 0 = 2 (x − √3)
2x − y + 2√3 = 0
Question 2 :
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and it passes through the point (5,3). Find the co-ordinates of the point A.
Let us represent the given details in a rough diagram.
The reflection point of A is A'. Its coordinate will be (1, -2).
Equation of A'B :
A'(1, -2) B (5, 3)
(y - y1) / (y2 - y1) = (x - x1)/(x2 - x1)
(y - 3) / (3 + 2) = (x - 5)/(5 - 1)
(y - 3) / 5 = (x - 5)/4
4 (y - 3) = 5 (x - 5)
4y - 12 = 5x - 25
5x - 4y - 25 + 12 = 0
5x - 4y - 13 = 0
The required point (x, 0) lies on the line A'B
5x - 4(0) - 13 = 0
5x - 13 = 0
5x = 13
x = 13/5
Hence the required point is (13/5, 0).
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