###
**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.

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 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.

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: Slope of a**

Straight Line

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

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.

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-

y1) and B (x2, y2). 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

y1 and the vertical distance between B and x-axis is y2,

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

distance between the point B and P= y2 – y1

Similarly,

the distance between the point A and P= x2 – x1

the distance between the point A and P= x2 – x1

We

know the formula for tan θ-

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-

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

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.

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

Line: General Form

The

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

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

**Equation of a Straight**

Line: Equation of x-Axis

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-

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.

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-

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.

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

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-

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.

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-

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.

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

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-

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.

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-

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.

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

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 –

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.

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-

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.

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

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.

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.

‘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

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.

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

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.

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

Line: Intercept Form

In

this form of the equation, intercepts are on both- the x-axis and the y-axis

are given.

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.

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

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.

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

us start by taking two lines

A

given pair of lines is called

i.e. they lie on top of each other. So, two lines are coincident if-

given pair of lines is called

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

A

given pair of lines is called

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-

given pair of lines is called

**if both the have the same***Parallel*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-

A

given pair of lines is called

at an angle of 900 and the product of their slopes is -1. So, two

lines are perpendicular if-

given pair of lines is called

**if both the lines are***Perpendicular*at an angle of 900 and the product of their slopes is -1. So, two

lines are perpendicular if-

###

**Co-ordinate Geometry: Distance**

of a Point from a Line

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-

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-

The

distance between two parallel lines, ax + by+ c1= 0 and ax +by + c2=0,

is calculated by using the formula-

distance between two parallel lines, ax + by+ c1= 0 and ax +by + c2=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.

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

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.

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

The points (2, 7) with a slope of 1 unit

(ii)

The points (5, 3) and (-2, 6)

The points (5, 3) and (-2, 6)

**Solution 1 (i):**

We

know the Slope point form of the Equation of a Line-

know the Slope point form of the Equation of a Line-

Using

the above, we can easily solve this question by substituting the values-

the above, we can easily solve this question by substituting the values-

y

– 7 = 1 (x- 2)

– 7 = 1 (x- 2)

x-

y + 5 = 0

y + 5 = 0

So

the equation for the line is x- y + 5 = 0

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.

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-

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-

the above, we can easily solve this question by substituting the values-

y

– 3 = [(6-3)/ (-2 -5) (x – 5)

– 3 = [(6-3)/ (-2 -5) (x – 5)

y

– 3 = (3/-7) (x -5)

– 3 = (3/-7) (x -5)

-7

(y – 3) = 3 (x -5)

(y – 3) = 3 (x -5)

3x

+ 7y -36 = 0

+ 7y -36 = 0

So the equation for the line is 3x + 7y -36 =

0

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.

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

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

Cuts only x-axis

(ii)

Cuts only y-axis

Cuts only y-axis

(iii)

Cuts both the axes

Cuts both the axes

(iv)

Does not cut any axis

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.

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.

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.

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.

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.

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

###
**Co-ordinate Geometry**

Problems Set 3: Find the Quadrant the Line Passes through

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

1st, 2nd and 3rd quadrant

(ii)

1st, 2nd and 4th quadrant

1st, 2nd and 4th quadrant

(iii)

2nd, 3rd and 4th quadrant

2nd, 3rd and 4th quadrant

(iv)

1st, 3rd and 4th quadrant

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-

of the ways of solving this question is of converting the straight line

equation in the intercept form. We get-

=>

4x + 3y = 12

4x + 3y = 12

=>

4x/12 + 3y/12 = 1

4x/12 + 3y/12 = 1

=>

x/3 + y/4 = 1

x/3 + y/4 = 1

From

this we can find that the x intercept is 3 and the y intercept is 4.

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 1st,

2nd and 4th quadrant.

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.

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

###
**Co-ordinate Geometry**

Problems Set 4: Parallel and Perpendicular Lines

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,

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.

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-

this we can say that the line whose equation we have to find will be of the

format-

4x

+ 5y + d = 0

+ 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

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

(4×4)

+ (5x-5) + d = 0

+ (5x-5) + d = 0

16

– 25 + d = 0

– 25 + d = 0

d

= 9

= 9

So

the equation of the parallel line will be => 4x + 5y + 9 = 0

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.

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.

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

Geometry

Question 1: Find

the equation of a straight line passing through the point (2,7) and having a

slope of 1 unit.

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.

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)

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

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

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

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

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

= 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.

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!

don’t forget to keep practicing!

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