Friday, 16 June 2017

Co-ordinate Geometry III- Different form of Equations of a Straight Line for SSC Exams

Co-ordinate Geometry is a vast topic and that covers the various forms of the equation of a straight line too. In this blog we will discuss the different forms of the equation of a straight line and questions on it.

Co-ordinate Geometry
Co-ordinate Geometry is a vast topic for SSC Exams, but if approached the right way it is extremely simple and scoring too. This is in our 3rd blog in this series of co-ordinate geometry where we have already discussed- the basics of Cartesian co-ordinate geometry, distance and section formula from co-ordinate geometry. Now we will discuss the various forms of the equation of a straight line. But before we move on to do that, a quick revision of the basics will give you a heads up.
Co-ordinate Geometry

Co-ordinate Geometry: Slope of a Straight Line

Co-ordinate Geometry
There are two mutually perpendicular axis, x and y, which intersect at the origin and there is a straight line at some angle of inclination. The first step here is to measure the slope of this line, how do we do it? Well, before we move to that remember slope is also called gradient, denoted by the letter ‘m’.
Co-ordinate Geometry
θ is the angle of inclination that the straight line made with the x-axis. The angle here is measured in the anti-clock wise direction from the positive x-axis.
Let us assume two points on the line- A (x1, y1) and B (x2, y2). In this case the slope of the line will be represented as-
Co-ordinate Geometry
Let’s see how we derived this equation. Look at the two points- A and B, the angle that this line makes with x-axis is θ only. We drop a perpendicular from B and extend a line from point A. The vertical distance between point A and x-axis is y1 and the vertical distance between B and x-axis is y2,
The distance between the point B and P= y2  - y1
Similarly, the distance between the point A and P= x2  - x1
We know the formula for tan θ-
Co-ordinate Geometry
In this case, the opposite side is BP and the adjacent side is AP, so we get the expression-
Co-ordinate Geometry


Co-ordinate Geometry: Equation of a Straight Line

A straight line on the xy plane can be represented by an equation, a relationship between x and y. This equation is satisfied by all the co-ordinates of all the points that lie on that line. This equation can be framed in various ways, which is what we’ll discuss in this section.

Equation of a Straight Line: General Form
Co-ordinate Geometry
The slope of the line can be determined from this equation of a straight line.
Co-ordinate Geometry

Equation of a Straight Line: Equation of x-Axis
The x-axis is also a straight line, therefore let’s see what will be the equation for x-axis-
Co-ordinate Geometry
It is so because, any point that lies on this line, the y co-ordinate for that will always be equal to zero.
Co-ordinate Geometry
So what about the slope here? Since slope is –a/b. In this equation there is no x variable at all, so-
Co-ordinate Geometry

Also we know that slope of a line is tanθ, but there the angle of x-axis with itself is 0 and tan0 is 0.

Equation of a Straight Line: Equation of y-Axis
The y-axis is also a straight line, therefore let’s see what will be the equation for y-axis-
Co-ordinate Geometry
It is so because, any point that lies on this line, the x co-ordinate for that will always be equal to zero.
Co-ordinate Geometry
So what about the slope here? Since slope is –a/b. In this equation there is no y variable at all, so-
Co-ordinate Geometry
Also we know that slope of a line is tanθ, but there the angle of y-axis with x-axis is 90 and tan90 is undefined.

Equation of a Straight Line: Equation of a Line Parallel to x-Axis
A line that is parallel to the x-axis will never meet the x-axis, the distance between the x-axis and the line will be consistent. So the equation here will be-
Co-ordinate Geometry
It is so because, any point that lies on this line, the y co-ordinate for that will always be equal to zero.
Co-ordinate Geometry
So what about the slope here? Since slope is –a/b. In this equation there is no x variable at all, so-
Co-ordinate Geometry
Also we know that slope of a line is tanθ, but there the angle of x-axis with itself is 0 and tan0 is 0.

Equation of a Straight Line: Equation of a Line Parallel to y-Axis
A line that is parallel to the y-axis will never meet the y-axis, the distance between the y-axis and the line will be consistent. So the equation here will be -
Co-ordinate Geometry
It is so because, any point that lies on this line, the y co-ordinate for that will always be equal to zero.
Co-ordinate Geometry
So what about the slope here? Since slope is –a/b. In this equation there is no y variable at all, so-
Co-ordinate Geometry
Also we know that slope of a line is tanθ, but there the angle of y-axis with x-axis is 90 and tan90 is undefined.

Equation of a Straight Line: Slope Intercept Form
In this form of the equation of a straight line in co-ordinate geometry we use the y intercept and the slope of the line to frame the equation.
Co-ordinate Geometry
Co-ordinate Geometry

Here ‘m’ is the slope of the line and ‘c’ is the intercept that is cut off by the y-axis on the line. Now you must be wondering what is an intercept? The intercept is the distance between the origin and the point where the line cuts the y-axis. Here when the value of ‘c’ is zero, it implies that the line passes through the origin.

Equation of a Straight Line: Slope Point Form
In this form of the equation of a straight line in co-ordinate geometry we use the co-ordinates of a point through which the line passes and the slope of the line to frame the equation.
Co-ordinate Geometry

Co-ordinate Geometry


Equation of a Straight Line: Two Point Form
As the name suggests, in this from of the equation two sets of co-ordinate points are given through which the line passes.
Co-ordinate Geometry

Co-ordinate Geometry

Equation of a Straight Line: Intercept Form
In this form of the equation, intercepts are on both- the x-axis and the y-axis are given.
Co-ordinate Geometry
Co-ordinate Geometry

These are all the different ways in which the equation of a line can be written in co-ordinate geometry.


Co-ordinate Geometry: Conditions for Parallel and Perpendicular Lines

These given conditions in co-ordinate geometry, will help us determine if the given pair of line are parallel or perpendicular.
Let us start by taking two lines
Co-ordinate Geometry
A given pair of lines is called Co-incident if both the lines coincide, i.e. they lie on top of each other. So, two lines are coincident if-
Co-ordinate Geometry
A given pair of lines is called Parallel if both the have the same slope, i.e. they have the same inclination with respect to the positive x-axis in the anti-clockwise direction. So, two lines are parallel if-
Co-ordinate Geometry
A given pair of lines is called Perpendicular if both the lines are at an angle of 900 and the product of their slopes is -1. So, two lines are perpendicular if-
Co-ordinate Geometry


Co-ordinate Geometry: Distance of a Point from a Line

The length of a perpendicular, or a straight line, from a point A (x1, y1) to a line ax + by + c=0, is calculated by using the formula-
Co-ordinate Geometry
The distance between two parallel lines, ax + by+ c1= 0 and ax +by + c2=0, is calculated by using the formula-
Co-ordinate Geometry
Now that we have all concepts in co-ordinate geometry about the equation of a straight line, let’s move on to questions based on them.


Co-ordinate Geometry Problems Set 1: Frame the Equation of a line

In such questions, conditions are given and based on the given conditions, you have to frame the equation of the line.

Example 1: Find the equation of a line passing through
(i) The points (2, 7) with a slope of 1 unit
(ii) The points (5, 3) and (-2, 6)

Solution 1 (i):
We know the Slope point form of the Equation of a Line-
Co-ordinate Geometry
Using the above, we can easily solve this question by substituting the values-
y – 7 = 1 (x- 2)
x- y + 5 = 0
So the equation for the line is x- y + 5 = 0
Another way of solving such questions in SSC Exams would be by substitution. There would be option, substitute the values of x and y as 2 and 7 respectively. The equation that will satisfy these numbers and whose slope is 1, will be the correct answer.

Solution 1 (ii):

We know the formula for the equation of a line when two points on a line are given-
Co-ordinate Geometry
Using the above, we can easily solve this question by substituting the values-
y – 3 = [(6-3)/ (-2 -5) (x – 5)
y – 3 = (3/-7) (x -5)
-7 (y – 3) = 3 (x -5)
3x + 7y -36 = 0
 So the equation for the line is 3x + 7y -36 = 0
Another way of solving such questions in SSC Exams would be by substitution. There would be option, substitute the values of x1, x2, y1 and y2. The equation that will satisfy these numbers will be the correct answer.

Co-ordinate Geometry Problems Set 2: Find the Axis the Line Intersects

Example 1: A line passes through the points (-2, 8) and (5, 7). Which of the following is true?
(i) Cuts only x-axis
(ii) Cuts only y-axis
(iii) Cuts both the axes
(iv) Does not cut any axis

Solution 1:
Now, the minute you read this question you will be tempted to quickly use the two point form of the equation of a line and solve it! But wait… this question doesn’t even need you to do that. Also option (iv) can easily be eliminated because no line that is drawn on the xy plane will not pass through either of the axes. Any line of the xy plane will atleast pass through one axis, else that line is not possible.
Let’s now plot these two lines on a graph.
Co-ordinate Geometry
The minute we plot the points on the graph and then draw a line to join them, we can see that it passes through the y-axis. So now, the next point to check is if it passes through the x-axis also. Looking at the line, we know that at some point it will pass through the x-axis because of the slope of the line. So we know that the line will pass through both- the x-axis and the y-axis.
There is a smart way to approach this question, without even plotting the graph. From the given co-ordinates we know that the given lines are neither parallel to the x-axis nor to the y-axis. The line that never meets y-axis will be parallel to y axis and the line that never meets x-axis will be parallel to x-axis. If the line is not parallel to either of the axis, the line is bound to pass by both the axes at some point or the other.

Therefore the answer is option (iii), it cuts both the axes.

Co-ordinate Geometry Problems Set 3: Find the Quadrant the Line Passes through

Example 1: The straight => 4x + 3y = 12, passes through which of the following quadrants?
(i) 1st, 2nd and 3rd quadrant
(ii) 1st, 2nd and 4th quadrant
(iii) 2nd, 3rd and 4th quadrant
(iv) 1st, 3rd and 4th quadrant

Solution 1:
One of the ways of solving this question is of converting the straight line equation in the intercept form. We get-
=> 4x + 3y = 12
=> 4x/12 + 3y/12 = 1
=> x/3 + y/4 = 1
From this we can find that the x intercept is 3 and the y intercept is 4.
Co-ordinate Geometry
From the above graph we can easily conclude that the line passes through 1st, 2nd and 4th quadrant.

Therefore the answer is option (ii), 1st, 2nd and 4th quadrant.

Co-ordinate Geometry Problems Set 4: Parallel and Perpendicular Lines

Example 1: What is the equation of a line which is parallel to => 4x + 5y = 18, and passes through the points (4, -5)?

Solution 1:

We know two line are parallel if they are in the form=>  ax + by + c = 0 and ax + by + d = 0,
This implies that the co-efficients have to be the same, but the value of constants varies.

Based on this we can say that the line whose equation we have to find will be of the format-
4x + 5y + d = 0

We know the parallel line passes through the points (4, -5), so we can simply substitute the values of x and u and get the value of d. So substituting values we get
(4x4) + (5x-5) + d = 0
16 – 25 + d = 0
d = 9
So the equation of the parallel line will be => 4x + 5y + 9 = 0

The smart way to solve this question in SSC Exams would be eliminating the options that are given. 

We know the slope of parallel lines is the same and in this case it should be => -4/5. You can eliminate based on this and then substitute the values of x and y to see which equation satisfies the given condition.

Practice Problems in Co-ordinate Geometry

Question 1: Find the equation of a straight line passing through the point (2,7) and having a slope of 1 unit.
a) x-y + 5=0      b) x+y-5=0      c) x+y+5=0      d) x –y-5=0 15.

Question 2: Find the equation of a straight line passing through the points (5,3) and (-2,6)
a) 3x-7y+36=0      b) 3x+7y-36=0      c) 3x+7y+36=0      d) 3x – 7y-36=0

Question 3: The equation of a line passing through (0,0) and parallel to the straight line 3x - 4y - 7=0, is
a) 4y - 3x = 0      b) 3x + y =0      c) 3x – y =2      d) 3y - 2x = 1

Question 4: Equation of the straight line parallel to x-axis and also 3 units below x –axis is
a) x = -3      b) y = 3      c) y = -3      d) x = 3


Remember to write your answers in the comment section and stay tuned for the next blog in this series.
Till then don’t forget to keep practicing!
Co-ordinate Geometry
Co-ordinate Geometry
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