Coordinate Geometry is considered to be one of the most interesting concepts of mathematics. Coordinate Geometry (or the analytic geometry) describes the link between geometry and algebra through graphs involving curves and lines.

It provides geometric aspects in Algebra and enables them to solve geometric problems. It is a part of geometry where the position of points on the plane is described using an ordered pair of numbers.

Here, the concepts of coordinate geometry (also known as Cartesian geometry) are explained along with its formulas and their derivations.

Introduction to Coordinate Geometry

Coordinate geometry (or analytic geometry) is defined as the study of geometry using the coordinate points. Using coordinate geometry, it is possible to find the distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc. There are certain terms in Cartesian geometry that should be properly understood. These terms include:

What is a Co-ordinate and a Co-ordinate Plane?

You must be familiar with plotting graphs on a plane, from the tables of numbers for both linear and non-linear equations. The number line which is also known as a Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis (horizontal line) and the y-axis(vertical line).

The four quadrants along with their respective values are represented in the graph below-

• Quadrant 1 : (+x, +y)
• Quadrant 2 : (-x, +y)
• Quadrant 3 : (-x, -y)
• Quadrant 4 : (+x, -y)

The point at which the axes intersect is known as the origin. The location of any point on a plane is expressed by a pair of values (x, y) and these pairs are known as the coordinates.

The figure below shows the Cartesian plane with coordinates (4,2). If the coordinates are identified, the distance between the two points and the interval’s midpoint that is connecting the points can be computed.

Coordinate Geometry Fig. 1: Cartesian Plane

Equation of a Line in Cartesian Plane

Equation of a line can be represented in many ways, few of which is given below-

(i) General Form

The general form of a line is given as Ax + By + C = 0.

(ii) Slope intercept Form 

Let x, y be the coordinate of a point through which a line passes, m be the slope of a line, and c be the y-intercept, then the equation of a line is given by:

y=mx + c

(iii) Intercept Form of a Line

Consider a and b be the x-intercept and y-intercept respectively, of a line, then the equation of a line is represented as-

y = mx + c

Slope of a Line: 

Consider the general form of a line Ax + By + C = 0, the slope can be found by converting this form to the slope-intercept form.

Ax + By + C = 0

⇒ By = − Ax – C

or,

Comparing the above equation with y = mx + c,

Thus, we can directly find the slope of a line from the general equation of a line.

Coordinate Geometry Formulas and Theorems

Distance Formula: To Calculate Distance Between Two Points

Let the two points be A and B, having coordinates to be (x₁, y₁) and (x₂, y₂), respectively.

Thus, the distance between two points is given as-

Coordinate Geometry Fig. 2: Distance Formula

Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points

Consider the same points A and B, which have coordinates (x₁, y₁) and (x₂, y₂), respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of point M is given as-

Angle Formula: To Find The Angle Between Two Lines

Consider two lines A and B, having their slopes m₁ and m_2, respectively.

Let “θ” be the angle between these two lines, then the angle between them can be represented as-

Special Cases:

• Case 1: When the two lines are parallel to each other,

m₁ = m₂ = m

Substituting the value in the equation above,

• Case 2: When the two lines are perpendicular to each other,

m₁ . m₂ = -1

Substituting the value in the original equation,

which is undefined.

⇒ θ = 90°

Section Formula: To Find a Point Which Divides a Line into m:n Ratio

Consider a line A and B having coordinates (x₁, y₁) and (x₂, y₂), respectively. Let P be a point that which divides the line in the ratio m:n, then the coordinates of the coordinates of the point P is given as-

• When the ratio m:n is internal:

• When the ratio m:n is external:

Students can follow the link provided to learn more about the section formula along its proof and solved examples.

Area of a Triangle in Cartesian Plane

The area of a triangle In coordinate geometrywhose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

If the area of a triangle whose vertices are (x₁, y₁),(x₂, y₂) and (x₃, y₃) is zero, then the three points are collinear.

• Important: Click here to Download Co-ordinate Geometry pdf

Examples Based On Coordinate Geometry Concepts

Examples 1: Find the distance between points M (4,5) and N (-3,8).

Solution:

Applying the distance formula we have,

Example 2: Find the equation of a line parallel to 3x+4y = 5 and passing through points (1,1).

Solution:

For a line parallel to the given line, the slope will be of the same magnitude.

Thus the equation of a line will be represented as 3x+4y=k

Substituting the given points in this new equation, we have

k = 3 × 1 + 4 × 1 = 3 + 4 = 7

Therefore the equation is 3x + 4y = 7

Coordinate Geometry Questions For Practice

1. Calculate the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).
2. Find the area of the triangle having vertices at A, B, and C which are at points (2, 3), (–1, 0), and (2, – 4), respectively. Also, mention the type of triangle.
3. A point A is equidistant from B(3, 8) and C(-10, x). Find the value for x and the distance BC.

Video Lesson on Coordinate Geometry Toughest Problems

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Conic Sections

Conic Section: a section (or slice) through a cone.

Cones

Circle
straight through

Ellipse
slight angle

Parabola
parallel to edge
of cone

Hyperbola
steep angle

So all those curves are related!

Focus!

The curves can also be defined using a straight line and a point (called the directrix and focus).

When we measure the distance:

• from the focus to a point on the curve, and
• perpendicularly from the directrix to that point

the two distances will always be the same ratio.

• For an ellipse, the ratio is less than 1
• For a parabola, the ratio is 1, so the two distances are equal.
• For a hyperbola, the ratio is greater than 1

Eccentricity

That ratio above is called the “eccentricity“, so we can say that any conic section is:

For:

• 0 < eccentricity < 1 we get an ellipse,
• eccentricity = 1 a parabola, and
• eccentricity > 1 a hyperbola.

A circle has an eccentricity of zero, so the eccentricity shows us how “un-circular” the curve is. The bigger the eccentricity, the less curved it is.

Latus Rectum

The latus rectum (no, it is not a rude word!) runs parallel to the directrix and passes through the focus. Its length:

• In a parabola, is four times the focal length
• In a circle, is the diameter
• In an ellipse, is 2b²/a (where a and b are one half of the major and minor diameter).

Here is the major axis and minor axis of an ellipse.

There is a focus and directrix on each side (ie a pair of them).

Equations

When placed like this on an x-y graph, the equation for an ellipse is:

x²a² + y²b² = 1

The special case of a circle (where radius=a=b):

x²a² + y²a² = 1

And for a hyperbola it is:

x²a² − y²b² = 1

General Equation

We can make an equation that covers all these curves.

Because they are plane curves (even though cut out of the solid) we only have to deal with Cartesian (“x” and “y”) Coordinates.

But these are not straight lines, so just “x” and “y” will not do … we need to go to the next level, and have:

• x² and y²,
• and also x (without y), y (without x),
• x and y together (xy)
• and a constant term.

There, that should do it!

And each one needs a factor (A,B,C etc) …

So the general equation that covers all conic sections is:

And from that equation we can create equations for the circle, ellipse, parabola and hyperbola.

Ellipse

An ellipse usually looks like a squashed circle:

The total distance from F to P to G stays the same

In other words, we always travel the same distance when going from:

• point “F” to
• to any point on the ellipse
• and then on to point “G”

You Can Draw It Yourself

Put two pins in a board, and then …

put a loop of string around them,

insert a pencil into the loop,

stretch the string so it forms a triangle,

and draw a curve.
It is an ellipse!

It works because the string naturally forces the same distance from pin-to-pencil-to-other-pin.

A Circle is an Ellipse

In fact a Circle is an Ellipse, where both foci are at the same point (the center).

In other words, a circle is a “special case” of an ellipse. Ellipses Rule!

Definition

Major and Minor Axes

The Major Axis is the longest diameter. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. And the Minor Axis is the shortest diameter (at the narrowest part of the ellipse).

The Semi-major Axis is half of the Major Axis, and the Semi-minor Axis is half of the Minor Axis.

Major Axis Equals f+g

Remember from the top how the distance “f+g” stays the same for an ellipse?

Well f+g is equal to the length of the major axis.

Can you think why? (Try moving the point P at the top.)

Calculations

Area is easy, perimeter is not!

Area

The area of an ellipse is:

π × a × b

where a is the length of the Semi-major Axis, and b is the length of the Semi-minor Axis.

Be careful: a and b are from the center outwards (not all the way across).

(Note: for a circle, a and b are equal to the radius, and you get π × r × r = πr², which is right!)

Perimeter Approximation

Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject: read Perimeter of an Ellipse for more details.

But a simple approximation that is within about 5% of the true value (so long as a is not more than 3 times longer than b) is as follows:

Remember, this is only a rough approximation! (That is why the “equals sign” is squiggly.)

Tangent

A tangent line just touches a curve at one point, without cutting across it. Here is a tangent to an ellipse:

Here is a cool thing: the tangent line has equal angles with the two lines going to each focus! Try bringing the two focus points together (so the ellipse is a circle) … what do you notice?

Reflection

Light or sound starting at one focus point reflects to the other focus point (because angle in matches angle out):

Have a play with a simple computer model of reflection inside an ellipse.

Eccentricity

The eccentricity is a measure of how “un-round” the ellipse is.

Eccentricity

Eccentricity: how much a conic section (a circle, ellipse, parabola or hyperbola)
varies from being circular.

Different values of eccentricity make different curves:

• At eccentricity = 0 we get a circle
• for 0 < eccentricity < 1 we get an ellipse
• for eccentricity = 1 we get a parabola
• for eccentricity > 1 we get a hyperbola
• for infinite eccentricity we get a line

Eccentricity is often shown as the letter e (don’t confuse this with Euler’s number “e”, they are totally different)

Calculating The Value

	For a circle, eccentricity is 0
	For an ellipse, eccentricity is:
	For a parabola, eccentricity is 1
	For a hyperbola, eccentricity is:

The formula (using semi-major and semi-minor axis) is:

√(a²−b²)a

Section of a Cone

You can also get an ellipse when you slice through a cone (but not too steep a slice, or you get a parabola or hyperbola).

In fact the ellipse is a conic section (a section of a cone) with an eccentricity between 0 and 1.

Equation

By placing an ellipse on an x-y graph (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve is:

x²a² + y²b² = 1

(similar to the equation of the hyperbola: x²/a² − y²/b² = 1, except for a “+” instead of a “−”)

Or we can “parametric equations”, where we have another variable “t” and we calculate x and y from it, like this:

• x = a cos(t)
• y = b sin(t)

(Just imagine “t” going from 0° to 360°, what x and y values would we get?)

Parabola

When you kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it arcs up into the air and comes down again … … following the path of a parabola! (Except for how the air affects it.)

Definition

A parabola is a curve where any point is at an equal distance from:

• a fixed point (the focus ), and
• a fixed straight line (the directrix )

Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).

Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line.

Keep going until you have lots of little dots, then join the little dots and you will have a parabola!

Names

Here are the important names:

• the directrix and focus (explained above)
• the axis of symmetry (goes through the focus, at right angles to the directrix)
• the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

Reflector

And a parabola has this amazing property:

Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.

And that explains why that dot is called the focus …

… because that’s where all the rays get focused!

So the parabola can be used for:

• satellite dishes,
• radar dishes,
• concentrating the sun’s rays to make a hot spot,
• the reflector on spotlights and torches,
• etc

We also get a parabola when we slice through a cone (the slice must be parallel to the side of the cone).So the parabola is a conic section (a section of a cone).

Equations

The simplest equation for a parabola is y = x²

Turned on its side it becomes y² = x

(or y = √x for just the top half)

A little more generally:

y² = 4ax

where a is the distance from the origin to the focus (and also from the origin to directrix)

The equations of parabolas in different orientations are as follows:

Measurements for a Parabolic Dish

If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurements do you need?

To make it easy to build, let’s have it pointing upwards, and so we choose the x² = 4ay equation.

And we want “a” to be 200, so the equation becomes:

x² = 4ay = 4 × 200 × y = 800y

Rearranging so we can calculate heights:

y = x²/800

And here are some height measurements as you run along:

	Distance Along (“x”)	Height (“y”)
	0 mm	0.0 mm
	100 mm	12.5 mm
	200 mm	50.0 mm
	300 mm	112.5 mm
	400 mm	200.0 mm
	500 mm	312.5 mm
	600 mm	450.0 mm

Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus.

Hyperbola

Did you know that the orbit of a spacecraft can sometimes be a hyperbola?

A spacecraft can use the gravity of a planet to alter its path and propel it at high speed away from the planet and back out into space using a technique called “gravitational slingshot”.

If this happens, then the path of the spacecraft is a hyperbola.

(Play with this at Gravity Freeplay)

Definition

A hyperbola is two curves that are like infinite bows.

Looking at just one of the curves:

any point P is closer to F than to G by some constant amount

The other curve is a mirror image, and is closer to G than to F.

In other words, the distance from P to F is always less than the distance P to G by some constant amount. (And for the other curve P to G is always less than P to F by that constant amount.)

Each bow is called a branch and F and G are each called a focus.

Have a try yourself:

Try moving point P: what do you notice about the lengths PF and PG ?

Also try putting point P on the other branch.

There are some other interesting things, too:

On the diagram you can see:

• an axis of symmetry (that goes through each focus)
• two vertices (where each curve makes its sharpest turn)
• the distance between the vertices (2a on the diagram) is the constant difference between the lengths PF and PG
• two asymptotes which are not part of the hyperbola but show where the curve would go if continued indefinitely in each of the four directions

And, strictly speaking, there is also another axis of symmetry that goes down the middle and separates the two branches of the hyperbola.

Conic Section You can also get a hyperbola when you slice through a double cone. The slice must be steeper than that for a parabola, but does not have to be parallel to the cone’s axis for the hyperbola to be symmetrical. So the hyperbola is a conic section (a section of a cone).

Equation

By placing a hyperbola on an x-y graph (centered over the x-axis and y-axis), the equation of the curve is:

x²a² − y²b² = 1

Also:

One vertex is at (a, 0), and the other is at (−a, 0)

The asymptotes are the straight lines:

• y = (b/a)x
• y = −(b/a)x

(Note: the equation is similar to the equation of the ellipse: x²/a² + y²/b² = 1, except for a “−” instead of a “+”)