Point-slope Form

The equation of a line in point-slope form we will learn how to find the equation of the straight line which is inclined at a given angle to the positive direction of x-axis in anticlockwise sense and passes through a given point.

Let the line MN makes an angle θ with the positive direction of x-axis in anticlockwise sense and passes through the point Q (x1, y1). We have to find the equation of the line MN.

Let P (x, y) be any point on the line MN. But Q (x1, y1) is also a point on the same line. Therefore, the slope of the line MN = yy1xx1

Again, the line MN makes an angle θ with the positive direction of the axis of x; hence, the slope of the line = tan θ = m (say).

Therefore, yy1xx1 = m  

⇒ y - y1 = m (x - x1)

The above equation y - y1 = m (x - x1) is satisfied by the co-ordinates of any point P lying on the line MN.

Therefore, y - y1 = m (x - x1) represent the equation of the straight line AB.


Solved examples to find the equation of a line in point-slope form:

1. Find the equation of a straight line passing through (-9, 5) and inclined at an angle of 120° with the positive direction of x-axis.

Solution:

First find the slope of the line:

Here slope of the line (m) = tan 120° = tan (90° + 30°) = cot 30° = √3.

Given point (x1, y1) ≡ (-9, 5)

Therefore, x1 = -9 and y1 = 5

We know that the equation of a straight line passes through a given point (x1, y1) and has the slope ‘m’ is y - y1 = m (x - x1).

Therefore, the required equation of the straight lien is y - y1 = m (x - x1)

⇒ y - 5 = √3{x - (-9)}

⇒ y - 5 = √3(x + 9)

⇒ y - 5 = √3x + 9√3

⇒ √3x + 9√3 = y - 5

⇒ √3x - y + 9√3 + 5 = 0

 

2. A straight line passes through the point (2, -3) and makes an angle 135° with the positive direction of the x-axis. Find the equation of the straight line.

Solution:  

The required line makes an angle 135° with the positive direction of the axis of x.

Therefore, the slope of the required line = m= tan 135° = tan (90° + 45°) = - cot 45° = -1.

 Again, the required line passes through the point (2, -3).

We know that the equation of a straight line passes through a given point (x1, y1) and has the slope ‘m’ is y - y1 = m (x - x1).

Therefore, the equation of the required straight line is y - (-3) = -1(x -2)

⇒ y + 3 = -x + 2

⇒ x + y + 1 = 0


Notes:

(i) The equation of a straight line of the form y - y1 = m (x - x1) is called its point-slope form.

(ii) The equation of the line y - y1 = m (x - x1) is sometimes expressed in the following form:

y - y1 = m(x - x1)

We know that m = tan θ = sinθcosθ

⇒ y - y1 = sinθcosθ (x - x1)

xx1cosθ = yy1sinθ  = r, Where r = (xx1)2+(yy1)2 i.e., the distance between the points (x, y) and (x1, y1).

The equation of a straight line as xx1cosθ = yy1sinθ = r is called its Symmetrical form.

 The Straight Line




11 and 12 Grade Math

From Point-slope Form to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Area of a Square and Rectangle | Area of Squares & Rectan

    Jul 19, 25 05:00 AM

    Area and Perimeter of Square and Rectangle
    We will practice the questions given in the worksheet on area of a square and rectangle. We know the amount of surface that a plane figure covers is called its area. 1. Find the area of the square len…

    Read More

  2. Area of Rectangle Square and Triangle | Formulas| Area of Plane Shapes

    Jul 18, 25 10:38 AM

    Area of a Square of Side 1 cm
    Area of a closed plane figure is the amount of surface enclosed within its boundary. Look at the given figures. The shaded region of each figure denotes its area. The standard unit, generally used for…

    Read More

  3. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  4. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  5. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More