# Position of a Point with Respect to a Circle

We will learn how to find the position of a point with respect to a circle.

A point (x$$_{1}$$, y$$_{1}$$) lies outside, on or inside a circle S = x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 according as S$$_{1}$$ > = or <0, where S$$_{1}$$ = x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c.

Let P (x$$_{1}$$, y$$_{1}$$) be a given point, C (-g , -f) be the centre and a be the radius of the given circle.

We need to find the position of the point P (x$$_{1}$$, y$$_{1}$$) with respect to the circle S = x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0.

Now, CP = $$\mathrm{\sqrt{(x_{1} + g)^{2} + (y_{1} + f)^{2}}}$$

Therefore, the point

(i) P lies outside the circle x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 if CP > the radius of the circle.

i.e., $$\mathrm{\sqrt{(x_{1} + g)^{2} + (y_{1} + f)^{2}}}$$ > $$\mathrm{\sqrt{g^{2} + f^{2} - c}}$$

⇒ $$\mathrm{(x_{1} + g)^{2} + (y_{1} + f)^{2}}$$ > g$$^{2}$$ + f$$^{2}$$ - c

⇒ x$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + g$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2fy$$_{1}$$ + f$$^{2}$$ > g$$^{2}$$ + f$$^{2}$$ – c

⇒ x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c > 0

⇒ S$$_{1}$$ > 0, where S$$_{1}$$ = x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c.

(ii) P lies on the circle x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 if CP = 0.

i.e., $$\mathrm{\sqrt{(x_{1} + g)^{2} + (y_{1} + f)^{2}}}$$ = $$\mathrm{\sqrt{g^{2} + f^{2} - c}}$$

⇒ $$\mathrm{(x_{1} + g)^{2} + (y_{1} + f)^{2}}$$ = g$$^{2}$$ + f$$^{2}$$ - c

⇒ x$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + g$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2fy$$_{1}$$ + f$$^{2}$$ = g$$^{2}$$ + f$$^{2}$$ – c

⇒ x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c = 0

⇒ S$$_{1}$$ = 0, where S$$_{1}$$ = x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c.

(iii) P lies inside the circle x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 if CP < the radius of the circle.

i.e., $$\mathrm{\sqrt{(x_{1} + g)^{2} + (y_{1} + f)^{2}}}$$ < $$\mathrm{\sqrt{g^{2} + f^{2} - c}}$$

⇒ $$\mathrm{(x_{1} + g)^{2} + (y_{1} + f)^{2}}$$ < g$$^{2}$$ + f$$^{2}$$ - c

⇒ x$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + g$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2fy$$_{1}$$ + f$$^{2}$$ < g$$^{2}$$ + f$$^{2}$$ – c

⇒ x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c < 0

⇒ S$$_{1}$$ < 0, where S$$_{1}$$ = x$$_{1}$$$$^{2}$$ + y$$_{1}$$$$^{2}$$ + 2gx$$_{1}$$ + 2fy$$_{1}$$ + c.

Again, if the equation of the given circle be (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ then the coordinates of the centre C (h, k) and the radius of the circle = a

We need to find the position of the point P (x$$_{1}$$, y$$_{1}$$) with respect to the circle (x - h)$$^{2}$$ + (y - k)$$^{2}$$= a$$^{2}$$.

Therefore, the point

(i) P lies outside the circle (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ if CP > the radius of the circle

i.e., CP > a

⇒ CP$$^{2}$$ > a$$^{2}$$

⇒ (x$$_{1}$$ - h)$$^{2}$$ + (y$$_{1}$$ - k)$$^{2}$$ > a$$^{2}$$

(ii) P lies on the circle (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ if CP = the radius of the circle

i.e., CP = a

⇒ CP$$^{2}$$ = a$$^{2}$$

⇒ (x$$_{1}$$ - h)$$^{2}$$ + (y$$_{1}$$ - k)$$^{2}$$ = a$$^{2}$$

(iii) P lies inside the circle (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ if CP < the radius of the circle

i.e., CP < a

⇒ CP$$^{2}$$ < a$$^{2}$$

⇒ (x$$_{1}$$ - h)$$^{2}$$ + (y$$_{1}$$ - k)$$^{2}$$ < a$$^{2}$$

Solved examples to find the position of a point with respect to a given circle:

1. Prove that the point (1, - 1) lies within the circle x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 = 0, whereas the point (-1, 2) is outside the circle.

Solution:

We have x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 = 0 ⇒ S = 0, where S = x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4

For the point (1, -1), we have S$$_{1}$$ = 1$$^{2}$$ + (-1)$$^{2}$$ - 4 ∙1 + 6 ∙ (- 1) + 4 = 1 + 1 - 4 - 6 + 4 = - 4 < 0

For the point (-1, 2), we have S$$_{1}$$ = (- 1 )$$^{2}$$ + 2$$^{2}$$ - 4 ∙ (-1) +  6 ∙ 2 + 4 = 1 + 4 + 4 + 12 + 4 = 25 > 0

Therefore, the point (1, -1) lies inside the circle whereas (-1, 2) lies outside the circle.

2. Discuss the position of the points (0, 2) and (- 1, - 3) with respect to the circle x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 = 0.

Solution:

We have x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 = 0 ⇒ S = 0 where S = x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4

For the point (0, 2):

Putting x = 0 and y = 2 in the expression x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 we have,

S$$_{1}$$ = 0$$^{2}$$ + 2$$^{2}$$ - 4 ∙ 0 + 6 ∙ 2 + 4 = 0 + 4 – 0 + 12 + 4 = 20, which is positive.

Therefore, the point (0, 2) lies within the given circle.

For the point (- 1, - 3):

Putting x = -1 and y = -3 in the expression x$$^{2}$$ + y$$^{2}$$ - 4x + 6y + 4 we have,

S$$_{1}$$ = (- 1)$$^{2}$$ + (- 3)$$^{2}$$ - 4 ∙ (- 1) + 6 ∙ (- 3) + 4 = 1 + 9 + 4 - 18 + 4 = 18 - 18 = 0.

Therefore, the point (- 1, - 3) lies on the given circle.

The Circle

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Thousandths Place in Decimals | Decimal Place Value | Decimal Numbers

Jul 20, 24 03:45 PM

When we write a decimal number with three places, we are representing the thousandths place. Each part in the given figure represents one-thousandth of the whole. It is written as 1/1000. In the decim…

2. ### Hundredths Place in Decimals | Decimal Place Value | Decimal Number

Jul 20, 24 02:30 PM

When we write a decimal number with two places, we are representing the hundredths place. Let us take plane sheet which represents one whole. Now, we divide the sheet into 100 equal parts. Each part r…

3. ### Tenths Place in Decimals | Decimal Place Value | Decimal Numbers

Jul 20, 24 12:03 PM

The first place after the decimal point is tenths place which represents how many tenths are there in a number. Let us take a plane sheet which represents one whole. Now, divide the sheet into ten equ…

4. ### Representing Decimals on Number Line | Concept on Formation of Decimal

Jul 20, 24 10:38 AM

Representing decimals on number line shows the intervals between two integers which will help us to increase the basic concept on formation of decimal numbers.