Question

if a line joining two points (3,0)and(5,2) is rotated about the point (3,0) in counter clockwise direction through an angle 15degree than find the equation the line in the new position​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

A line joining two points (3,0)and(5,2) is rotated about the point (3,0) in counter clockwise direction through an angle 15°.

Let assume that the coordinates (3, 0) and (5, 2) is represented as A and B and let AB makes an angle p with the positive direction of x axis measured in anti-clockwise direction.

We know,

Slope of a line joining two points (a, b) and (c, d) and which makes an angle p with the positive direction of x axis measured in anti-clockwise direction is given by

$\rm :\longmapsto\:\boxed{ \tt{ \: tanp \: = \: \frac{d - b}{c - a} \: }}$

So, using this, we get

$\rm :\longmapsto\:tanp = \dfrac{2 - 0}{5 - 3}$

$\rm :\longmapsto\:tanp = \dfrac{2}{2}$

$\rm :\longmapsto\:tanp = 1$

$\rm :\longmapsto\:tanp = tan45 \degree$

$\bf\implies \:\:p = 45 \degree$

Now,

As it is given that,

AB rotated about the point A(3, 0) in clockwise direction through an angle of 15°

It means,

The required line makes an angle of 30° with the positive direction of x axis measured in anti-clockwise direction and passes through the point A( 3, 0 ).

We know, equation of line which passes through the point (a, b) and makes an angle p with positive direction of x axis measured in anti-clockwise direction is given by

$\rm :\longmapsto\:\boxed{ \tt{ \: y - b \: = \: tanp(x - a) \: }}$

So, required equation of line is

$\rm :\longmapsto\:y - 0 = tan30 \degree \:(x - 3)$

$\rm :\longmapsto\:y = \dfrac{1}{ \sqrt{3} } \:(x - 3)$

$\rm :\longmapsto\: \sqrt{3}y = x - 3$

$\rm \implies\:\boxed{ \tt{ \: x - \sqrt{3}y - 3 = 0 \: \: }}$

Additional Information :-

1. Equations of horizontal and vertical lines

Equation of the lines which is parallel to the X-axis is y = a, 

And

Equation of a straight line which is parallel to y - axis is x = a.

2. Point-slope form equation of line

The Equation of line which passes through the point (a, b) having slope m, is given by y - b = m(x - a)

3. Slope-intercept form

Equation of line which makes an intercept of c units on y - axis having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a units on b units on respective axis is given by x/a + y/b = 1.

5. Normal form of Line

Consider a line which is at a distance of p units from the origin and it makes an angle β with the positive x - axis.

Then, equation of line is given by x cosβ + y sinβ = p.

Answer 2

Answer:

$⟹x−3y−3=0</p><p>$