**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.

**Solution :**

Let us represent the given information in a picture.

Let A (2, 0) and B (3, 1) be the given points.

Slope of PQ = (y_{2} − y_{1}) / (x_{2} − x_{1})

= (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.

**Solution :**

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 - y_{1}) / (y_{2} - y_{1}) = (x - x_{1})/(x_{2} - x_{1})

(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).

After having gone through the stuff given above, we hope that the students would have understood "Find the Equation of the Line in the New Position".

Apart from the stuff given above, if you want to know more about "Find the Equation of the Line in the New Position".

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