Question

Find the equation of circle with center at \( (2, 5) \) and radius 5 units.

• A. \( x^2 + y^2 + 4x - 10y + 4 = 0 \)
• B. \( x^2 + y^2 - 4x - 10y + 4 = 0 \)
• C. \( x^2 + y^2 + 4x + 10y + 4 = 0 \)
• D. \( x^2 + y^2 + 4x - 10y - 4 = 0 \)

Hint:

The standard equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Expanding this gives the general form \( x^2 + y^2 + Dx + Ey + F = 0 \), where \( D = -2h \), \( E = -2k \), and \( F = h^2 + k^2 - r^2 \).

Answer 1

The Correct Option is B

Step 1: Recall the standard equation of a circle with center \( (h, k) \) and radius \( r \). The equation is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Step 2: Identify the given values for the center and radius. Center \( (h, k) = (2, 5) \) Radius \( r = 5 \)

Step 3: Substitute these values into the standard equation. \[ (x - 2)^2 + (y - 5)^2 = 5^2 \] \[ (x - 2)^2 + (y - 5)^2 = 25 \]

Step 4: Expand the squared binomials. \[ (x^2 - 2(x)(2) + 2^2) + (y^2 - 2(y)(5) + 5^2) = 25 \] \[ (x^2 - 4x + 4) + (y^2 - 10y + 25) = 25 \]

Step 5: Rearrange the terms to get the general form \( x^2 + y^2 + Dx + Ey + F = 0 \). \[ x^2 + y^2 - 4x - 10y + 4 + 25 = 25 \] Subtract 25 from both sides: \[ x^2 + y^2 - 4x - 10y + 4 + 25 - 25 = 0 \] \[ x^2 + y^2 - 4x - 10y + 4 = 0 \]

Step 6: Compare the resulting equation with the given options. The equation \( x^2 + y^2 - 4x - 10y + 4 = 0 \) matches option (B).