Question

Find the equation of the line which cuts off an
intercept 4 on the x-axis and makes an angle of 30°
with the y-axis​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

• Find the equation of the line which cuts off an intercept 4 on the x-axis and makes an angle of 30° with the y-axis.

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

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

• A line which makes an intercept 4 on the x-axis.

• makes an angle of 30° with the y-axis.

$\large\underline{\bold{To \: Find - }}$

• The equation of line.

$\large\underline{\bold{Concept \: Used- }}$

• The slope of line which makes an angle 'a' with positive direction of x - axis is given by, m = tana

• The equation of line having slope 'm' and makes an intercept of 'c' on x - axis is given by y = m(x - a).

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

Since,

• Line makes an angle of 30° with y - axis,

it implies,

• it makes an angle of 120° with positive direction of x - axis.

Therefore,

• slope of line (m) is given by

$\rm :\longmapsto\:m \: = \: tan120 \degree \:$

$\rm :\longmapsto\:m \: = \: tan(90\degree \: + 30\degree \:)$

$\rm :\longmapsto\:m \: = \: - \: cot30\degree \:$

$\rm :\longmapsto\:m \: = \: - \: \sqrt{3}$

Now,

we have,

• a line having slope - √3 and intercept on x - axis is 4,

So,

• Equation of line is given by

$\rm :\longmapsto\:y = - \sqrt{3} (x - 4)$

$\rm :\longmapsto\:y = - \sqrt{3} x + 4 \sqrt{3}$

$\rm :\longmapsto\: \sqrt{3} x + y - 4 \sqrt{3} = 0$

Additional Information

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