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Two circles, each of radius 5 units, are drawn in the coordinate plane with their
centers (0, 0) and (8, 0) respectively. How many points where both coordinates are
integers lie within the intersection of these circles, including its boundary?

Respuesta :

oladayoademosu

The points where both coordinates are integers lie within the intersection of these circles, including its boundary are 8 points.

To answer the question, we need to find the equation of the circles.

What is the equation of a circle?

The equation of a circle with center (h, k) and radius r is

(x - h)² + (y - k)² = r²

Equation of first circle

Given that the first circle has

  • center (0, 0) and
  • radius, r = 5,

Its equation is

(x - 0)² + (y - 0)² = 5²

x² + y² = 25  (1)

Equation of second circle

Given that the second circle has

  • center (8, 0) and
  • radius, r = 5,

Its equation is

(x - 8)² + (y - 0)² = 5²

x² - 16x + 64 + y² = 25  (2)

Point of intersection o the circles

To find the point of intersection of both circles, subtracting (1) from (2), we have

x² - 16x + 64 + y² = 25  (2)

-

x² + y² = 25  (1)

-16x + 64 = 0

-16x = -64

x = -64/-16

x = 4

Substituting x = 4 into (1), we have

x² + y² = 25

4² + y² = 25

16 + y² = 25

y² = 25 - 16

y² = 9

y = ±√9

y = ±3

So, the circles intersect at (4,-3) and (4, 3).

So, all the points that lie within their points of intersection where both coordinates are integers including their points of intersection are

  • (4, -3),
  • (3, -2),
  • (2, -1),
  • (1, 0),
  • (0, 1),
  • (1, 2),
  • (2, 3) and
  • (3, 4).

So, there are 8 points.

So, all the points where both coordinates are integers lie within the intersection of these circles, including its boundary are 8 points.

Learn more about points of intersection of circle here:

https://brainly.com/question/27893140

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