a circle pass through -5,0 3,0 2,4 .find the equation of the circle.detrmine whetherP(2,8) lies in side outside or in the circle

To find the equation of the circle passing through the points (-5,0), (3,0), and (2,4), we can use the general equation of a circle, which is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius.

Step 1: Find the center of the circle The center of the circle is the midpoint of the line segment joining the points (-5,0) and (3,0). The midpoint formula is ((x₁+x₂)/2, (y₁+y₂)/2). So, the center of the circle is ((-5+3)/2, (0+0)/2) = (-1,0).

Step 2: Find the radius of the circle The radius of the circle is the distance from the center of the circle to any of the points on the circle. We can use the distance formula, which is √((x₂-x₁)² + (y₂-y₁)²). So, the radius of the circle is √((2-(-1))² + (4-0)²) = √(9+16) = √25 = 5.

Step 3: Write the equation of the circle Substitute the center (-1,0) and the radius 5 into the general equation of a circle. The equation of the circle is (x-(-1))² + (y-0)² = 5², or (x+1)² + y² = 25.

Step 4: Determine whether the point P(2,8) lies inside, outside, or on the circle Substitute the point (2,8) into the equation of the circle. If the left side of the equation is less than 25, the point is inside the circle. If it's more than 25, the point is outside the circle. If it's equal to 25, the point is on the circle.

The left side of the equation is (2-(-1))² + (8-0)² = 9 + 64 = 73, which is more than 25. So, the point P(2,8) lies outside the circle.