WRITING EQUATIONS OF CIRCLES IN STANDARD FORM FROM GRAPHS

Standard form of equation of circle

(x - h)2 + (y - k)2  = r2

To write the equation of the circle in standard form from graphs, we need to mark two things in the graph.

(i)  Centre of the circle

(ii)  Any point lies on the circle

Let the center point be (h, k).

By finding the distance from center to the point lies on the circle, we may obtain the radius of the circle.

Example 1 :

Write the equation of the circle in standard form from the graph given below.

Solution :

From the given graph, first let us mark the center point. Here (-1, 2) is the center point

(h, k)  ==> (1, 2)

One of the points lies on the circle is (2, 2).

Let us find the distance between the points (1, 2) and (2, 2)

d = √(x2-x1)2 + (y2 - y1)2

d = √(2 - 1)2 + (2 - 2)2

d = √12 + 02

d = 1 unit

Standard equation of the circle

(x - h)2 + (y - k)2  = r2

(x - 1)2 + (y - 2)2  = 12

(x - 1)2 + (y - 2)2  = 1

Example 2 :

Write the equation of the circle in standard form from the graph given below.

Solution :

From the given graph, first let us mark the center point.

Here (-2, -3) is the center point.

(h, k)  ==> (-2, -3)

One of the points lies on the circle is (-2, -1)

Let us find the distance between the points (-2, -3) and (-2, -1)

d = √(x2-x1)2 + (y2 - y1)2

d = √(-2 - (-2))2 + (-1 - (-3))2

d = √(-2 + 2)2 + (-1 + 3)2

d = √02 + 22

d = √4

d = 2 units

Standard equation of the circle

(x - (-2))2 + (y - (-3))2  = 22

(x + 2)2 + (y + 3)2  =  22

(x + 2)2 + (y + 3)2  = 4

Example 3 :

Write the equation of the circle in standard form from the graph given below.

Solution :

From the given graph, first let us mark the center point.

Here (-1, -2) is the center point.

(h, k)  ==> (-1, -2)

One of the points lies on the circle is (-1, -7)

Let us find the distance between the points (-1, -2) and (-1, -7)

d = √(x2-x1)2 + (y2 - y1)2

d = √(-7 - (-2))2 + (-1 - (-1))2

d = √(-7 + 2)2 + (-1 + 1)2

d = √(-5)2 + 02

d = √25

d = 5 units

Standard equation of the circle

(x - (-1))2 + (y - (-2))2  = 52

(x + 1)2 + (y + 2)2  =  25

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