Solving Circle Equation for \(x^2 + y^2\) Value

We are given the equation of a circle:

\(x^2 + y^2 - 16x + 38y + 425 = 0\)

Our goal is to find the specific value of the expression \(x^2 + y^2\).

Rearranging the Circle Equation

To find the values of x and y, we can rewrite the given equation by completing the square for both the x and y terms.

Group the x terms and y terms:

\((x^2 - 16x) + (y^2 + 38y) + 425 = 0\)

Completing the Square

Now, we complete the square for the grouped terms:

• For the x terms (\(x^2 - 16x\)): Add and subtract \((\frac{-16}{2})^2 = (-8)^2 = 64\).

  \(x^2 - 16x = (x^2 - 16x + 64) - 64 = (x - 8)^2 - 64\)

• For the y terms (\(y^2 + 38y\)): Add and subtract \((\frac{38}{2})^2 = (19)^2 = 361\).

  \(y^2 + 38y = (y^2 + 38y + 361) - 361 = (y + 19)^2 - 361\)

Substitute these back into the original equation:

\((x - 8)^2 - 64 + (y + 19)^2 - 361 + 425 = 0\)

Combine the constant terms:

\((x - 8)^2 + (y + 19)^2 - 64 - 361 + 425 = 0\)

\((x - 8)^2 + (y + 19)^2 - 425 + 425 = 0\)

\((x - 8)^2 + (y + 19)^2 = 0\)

Determining the Values of x and y

The equation \((x - 8)^2 + (y + 19)^2 = 0\) signifies that the sum of two squared terms is zero. Since squares of real numbers are always non-negative (greater than or equal to zero), this equality can only hold true if both squared terms are individually equal to zero.

• \(x - 8 = 0 \implies x = 8\)
• \(y + 19 = 0 \implies y = -19\)

This means the equation represents a single point (8, -19).

Calculating the Value of x² + y²

Now, we substitute the determined values of x and y into the expression \(x^2 + y^2\):

\(x^2 + y^2 = (8)^2 + (-19)^2\)

\(x^2 + y^2 = 64 + 361\)

\(x^2 + y^2 = 425\)

Therefore, the value of \(x^2 + y^2\) is 425.