Explain why the point A(7;2) does not lie on the circle x^2+y^2-16=0 .

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The curve defined by the equation x^2 + y^2 - 16 = 0 can be written as

x^2 + y^2 - 16 = 0

=> (x - 0)^2 + (y - 0)^2 = 4^2

This is the equation of a circle with center at (0,0) and a radius of 4.

The distance of the point (7 , 2) from (0, 0) is sqrt( 7^2 + 2^2)

= sqrt( 49 + 4)

= sqrt 53

For the point to lie on the circle the distance should be 4.

As sqrt 53 is not equal to 4, the point does not lie on the circle.

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To verify if a point is located on a circle, it's coordinates must cancel the equation of the circle.

We'll replace x and y with the values of coordinates of the point A.

7^2 + 2^2 - 16 = 0

49 + 4 - 16 = 0

37 = 0 not true!

It is obvious that the point is not located on the circle.

To determine where exactly the point is found, we'll determine the power of the point.

The power of A with respect to circle C is:

p(A) = d^2 - r^2

d - is the distance from A to the center of the circle C.

We'll calculate d^2 = 7^2 + 2^2

d^2 = 49 + 4

d^2 = 53

p(A) = 53 - 16

p(A) = 37 > 0

Since the result of the power of the point A, with respect to the circle C, is positive, the point is found outside the circle C.

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