Question

Find the equations of the straight line passing through the point [1,2] and making an angle of 60° with the line √ 3x+y+2=0

Answer 1

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

Given that,

A line 'l' passing through the point (1, 2) and making an angle of 60° with the line √ 3x+y+2=0.

Let assume that slope of the line 'l' be m.

Now, Consider the line l',

$\rm \: \sqrt{3}x + y + 2 = 0$

$\rm \: y = - \sqrt{3}x - 2$

So, it implies slope of line l', say M is

$\rm \: M = - \sqrt{3}$

Now, we know that

If slope of two lines be m and M, then angle p between the two lines is given by

$\boxed{ \rm{ \:tanp \: = \: \bigg |\dfrac{M - m}{1 + Mm} \bigg| \: }}$

Now, as line l and l' having slope m and - √3 and angle between them is 60°,

So, on substituting the values, we get

$\rm \: tan60 \degree \: = \: \bigg |\dfrac{ - \sqrt{3} - m}{1 - \sqrt{3} m} \bigg|$

$\rm \: \sqrt{3} \: = \: \bigg |\dfrac{ - \sqrt{3} - m}{1 - \sqrt{3} m} \bigg|$

$\rm \: \pm \: \sqrt{3} \: = \: \dfrac{ - \sqrt{3} - m}{1 - \sqrt{3} m}$

So, it means

$\rm \: \sqrt{3} \: = \: \dfrac{ - \sqrt{3} - m}{1 - \sqrt{3} m} \\ \rm \: or \\ - \sqrt{3} \: = \: \dfrac{ - \sqrt{3} - m}{1 - \sqrt{3} m}$

$\rm \: \sqrt{3} - 3m = - \sqrt{3} - m \: \: or \: \: - \sqrt{3} + 3m = - \sqrt{3} - m$

$\rm \: 2m = 2\sqrt{3} \: \: or \: \: 4m = 0$

$\rm\implies \:\rm \: m = \sqrt{3} \: \: or \: \: m = 0$

So, equation of line l which passes through the point (1, 2) and having slope m = √3 is given by

$\rm \: y - 2 = \sqrt{3}(x - 1)$

$\rm \: y - 2 = \sqrt{3}x - \sqrt{3}$

$\rm\implies \:\rm \:\boxed{ \rm{ \: \sqrt{3}x - y + 2 - \sqrt{3} = 0 \: \: }}$

Also, equation of line l which passes through the point (1, 2) and having slope m = 0 is given by

$\rm \: y - 2 = 0(x - 1)$

$\rm\implies \:\boxed{ \rm{ \: \: y \: = \: 2 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

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

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.