Write the equation of the circle in standard form. Find the center, radius, intercepts, and graph the circle. x2+y2+16x−18y+145=25

Standard form (x-h)^2 +(y-k)^2 k

To determine the equation of the circle in standard form and find the center, radius, intercepts, and graph the circle, we'll start by rearranging the equation given:

x^2 + y^2 + 16x - 18y + 145 = 25

To begin, let's group the x and y terms and move the constant term to the other side of the equation:

x^2 + 16x + y^2 - 18y = 25 - 145

Simplifying further:

x^2 + 16x + y^2 - 18y = -120

Now, we can complete the square for both the x and y terms. To complete the square for the x terms, we need to add the square of half the coefficient of x to both sides of the equation. Similarly, for the y terms, we'll add the square of half the coefficient of y:

(x^2 + 16x + 64) + (y^2 - 18y + 81) = -120 + 64 + 81

Simplifying further:

(x + 8)^2 + (y - 9)^2 = 25

Now we have successfully completed the square for both the x and y terms. The equation is now in the standard form of a circle:

(x - h)^2 + (y - k)^2 = r^2

Comparing this equation to the one we obtained, we can conclude that the center of the circle is at the point (-8, 9) since h = -8 and k = 9. The radius can be determined from the equation where r^2 = 25, so the radius, r, is 5.

To find the intercepts of the circle, we can set y = 0 and solve for x and set x = 0 and solve for y. Let's start with setting y = 0:

(x + 8)^2 + (0 - 9)^2 = 25

(x + 8)^2 + 81 = 25

(x + 8)^2 = 25 - 81

(x + 8)^2 = -56

Since the square of a real number cannot be negative, this means there are no x-intercepts for this circle.

Now, let's set x = 0:

(0 + 8)^2 + (y - 9)^2 = 25

64 + (y - 9)^2 = 25

(y - 9)^2 = 25 - 64

(y - 9)^2 = -39

Similarly, since the square of a real number cannot be negative, there are no y-intercepts for this circle either.

To graph the circle, plot the center point (-8, 9) and draw a circle around it with a radius of 5 units.

So, the equation of the circle in standard form is (x + 8)^2 + (y - 9)^2 = 25. The center of the circle is at the point (-8, 9), the radius is 5 units, and the circle does not intersect either the x-axis or the y-axis.