Question

(ii) Find the equation of the line which makes an angle of 75° with x-axis and
cuts an intercept of length 3 on the positive direction of y-axis.​

Answer 1

Given :-

• A line which makes an angle of 75° with x-axis

• cuts an intercept of length 3 on the positive direction of y-axis.

To Find :-

• Equation of line.

Understanding the concept Used :-

1. Slope of a line :-

• Let us assume a line which makes an angle 'a' with positive direction of x - axis, then slope of line 'm' is given by m = tana.

2. Slope Intercept form :-

• Let us assume a line which makes an intercept of 'c' units on positive direction of y - axis and having slope 'm', then equation of line is given by y = mx + c.

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

Given that

• A line which makes an angle of 75° with x-axis,

So,

• Slope of line, m is given by

$\sf\: \: m \: = \: tan \: 75 \degree \:$

$\sf\: \: m \: = \: tan( \: 45\degree \: + 30\degree \:)$

$\sf\: m \: = \: \dfrac{tan45\degree \: + tan30\degree \:}{1 - tan45\degree \:tan30\degree \:}$

$\: \: \: \: \: \boxed{ \bf{ \because \: tan(x + y) = \dfrac{tanx + tany}{1 - tanx \: tany} }}$

$\sf\: m \: = \:\dfrac{1 + \dfrac{1}{ \sqrt{3} } }{1 - \: 1 \times \dfrac{1}{ \sqrt{3} } }$

$\sf\: m \: = \:\dfrac{ \sqrt{ 3} + 1}{ \sqrt{3} - 1 }$

$\sf\: m \: = \:\dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1} \times \dfrac{ \sqrt{3} + 1 }{ \sqrt{3} + 1 }$

$\sf\: m \: = \:\dfrac{ {( \sqrt{3} + 1) }^{2} }{ {( \sqrt{3} )}^{2} - {(1)}^{2} }$

$\sf\: m \: = \:\dfrac{3 + 1 + 2 \sqrt{3} }{3 - 1}$

$\sf\: m \: = \:\dfrac{4 + 2 \sqrt{3} }{2}$

$\boxed{ \bf{ \therefore \: m \: = \: 2 \: + \: \sqrt{3} }}$

Again,

Given that

• Line cuts an intercept of length 3 on the positive direction of y-axis.

$\sf\: \therefore \: c \: = \: 3$

Hence,

• The required equation of line using slope intercept form is given by

$\boxed{ \bf\: \: y \: = \: ( \: 2 \: + \: \sqrt{3} \: ) \: x \: + \: 3}$

Additional Information

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

• Equation of the lines which are horizontal or parallel to the X-axis is y = a, where a is the y – coordinate of the points on the line.

• Similarly, equation of a straight line which is vertical or parallel to Y-axis is x = a, where a is the x-coordinate of the points on the line

2. Point-slope form equation of line

• Consider a non-vertical line L whose slope is m, A(x,y) be an arbitrary point on the line and P(a, b) be the fixed point on the same line. Equation of line is given by y - b = m(x - a)

3. Slope-intercept form equation of line

• Consider a line whose slope is m which cuts the Y-axis at a distance ‘a’ from the origin. Then the distance a is called the y– intercept of the line. The point at which the line cuts y-axis will be (0,a). Then equation of line is given by y = mx + a.

4. Intercept Form of Line

• Consider a line L having x– intercept a and y– intercept b, then the line passes through  X– axis at (a,0) and Y– axis at (0,b). Equation of line is given by x/a + y/b = 1.

5. Normal form of Line

• Consider a perpendicular from the origin having length p to line L and it makes an angle β with the positive X-axis. Then, equation of line is given by x cosβ + y sinβ = p.