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




11 and 12 Grade Math 

From Position of a Point with Respect to a Circle to HOME PAGE




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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Writing Money in Words and Figure | Rules for Writing Money in Words

    Feb 11, 25 12:36 PM

    Rules for writing money in words and figure: 1. Abbreviation used for a rupee is Re. and for 1-rupee it is Re. 1 2. Rupees is written in short, as Rs., as 5-rupees is written as Rs. 5

    Read More

  2. Worksheet on Money | Conversion of Money from Rupees to Paisa

    Feb 11, 25 09:39 AM

    Amounts in Figures
    Practice the questions given in the worksheet on money. This sheet provides different types of questions where students need to express the amount of money in short form and long form

    Read More

  3. Worksheet on Measurement | Problems on Measurement | Homework |Answers

    Feb 10, 25 11:56 PM

    Measurement Worksheet
    In worksheet on measurement we will solve different types of questions on measurement of length, conversion of length, addition and subtraction of length, word problems on addition of length, word pro…

    Read More

  4. Worksheet on Subtraction of Capacity | Word Problems on Capacity | Ans

    Feb 10, 25 09:36 AM

    Subtraction of Volume Worksheet
    Practice the third grade math worksheet on subtraction of capacity. This sheet provides different types of questions where you need to arrange the values of capacity under

    Read More

  5. Practice Test on Circle | Quiz on Circle | Question and Test on Circle

    Feb 10, 25 09:08 AM

    Geometry practice test on circle, the questions we practiced and discussed under worksheets on circle are given here in geometry practice test.

    Read More