###
**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 is a vast topic for SSC Exams, but if approached the right way it is
extremely simple and scoring too. This is in our 3

^{rd}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: Slope of a
Straight Line **

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

**, denoted by the letter***gradient***.***‘m’*
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 (x

_{1, }y_{1}) and B (x_{2, }y_{2}). In this case the slope of the line will be represented as-
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
y

_{1}and the vertical distance between B and x-axis is y_{2},
The
distance between the point B and P= y

_{2 }- y_{1}
Similarly,
the distance between the point A and P= x

_{2 }- x_{1}
We
know the formula for tan θ-

In
this case, the opposite side is BP and the adjacent side is AP, so we get the
expression-

###
**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**

The
slope of the line can be determined from this equation of a straight line.

**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-

It
is so because, any point that lies on this line, the y co-ordinate for that
will always be equal to zero.

So
what about the slope here? Since slope is –a/b. In this equation there is no x variable
at all, so-

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-

It
is so because, any point that lies on this line, the x co-ordinate for that
will always be equal to zero.

So
what about the slope here? Since slope is –a/b. In this equation there is no y variable
at all, so-

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-

It
is so because, any point that lies on this line, the y co-ordinate for that
will always be equal to zero.

So
what about the slope here? Since slope is –a/b. In this equation there is no x variable
at all, so-

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 -

It
is so because, any point that lies on this line, the y co-ordinate for that
will always be equal to zero.

So
what about the slope here? Since slope is –a/b. In this equation there is no y variable
at all, so-

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.

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.

**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.

**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.

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

A
given pair of lines is called

**if both the lines coincide, i.e. they lie on top of each other. So, two lines are coincident if-***Co-incident*
A
given pair of lines is called

**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-***Parallel*
A
given pair of lines is called

**if both the lines are at an angle of 90***Perpendicular*^{0}and the product of their slopes is -1. So, two lines are perpendicular if-###

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

The
length of a perpendicular, or a straight line, from a point A (x

_{1}, y_{1}) to a line ax + by + c=0, is calculated by using the formula-
The
distance between two parallel lines, ax + by+ c

_{1}= 0 and ax +by + c_{2}=0, is calculated by using the formula-
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-

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-

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 x

_{1}, x_{2}, y_{1}and y_{2}. 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.

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)
1

^{st}, 2^{nd}and 3^{rd}quadrant
(ii)
1

^{st}, 2^{nd}and 4^{th}quadrant
(iii)
2

^{nd}, 3^{rd}and 4^{th}quadrant
(iv)
1

^{st}, 3^{rd}and 4^{th}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.

From
the above graph we can easily conclude that the line passes through 1

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

^{st}, 2^{nd}and 4^{th}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!

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