Convert the circle equation 4x2 + 4y2 - 6x + 10y - 1 = 0 to standard form and find its center and radius.

Step 1: Rewrite the equation in standard form. The given equation of the circle is 4x^2 + 4y^2 - 6x + 10y - 1 = 0. To convert this to the standard form (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius, first divide the entire equation by 4: x^2 + y^2 - (6)/(4)x + (10)/(4)y - (1)/(4) = 0 x^2 + y^2 - (3)/(2)x + (5)/(2)y - (1)/(4) = 0 Step 2: Group the x-terms and y-terms and move the constant to the right side. (x^2 - (3)/(2)x) + (y^2 + (5)/(2)y) = (1)/(4) Step 3: Complete the square for both the x-terms and y-terms. To complete the square for x^2 - (3)/(2)x, add ((1)/(2) · -(3)/(2))^2 = (-(3)/(4))^2 = (9)/(16). To complete the square for y^2 + (5)/(2)y, add ((1)/(2) · (5)/(2))^2 = ((5)/(4))^2 = (25)/(16). Add these values to both sides of the equation: (x^2 - (3)/(2)x + (9)/(16)) + (y^2 + (5)/(2)y + (25)/(16)) = (1)/(4) + (9)/(16) + (25)/(16) Step 4: Rewrite the expressions as squared terms and simplify the right side. (x - (3)/(4))^2 + (y + (5)/(4))^2 = (4)/(16) + (9)/(16) + (25)/(16) (x - (3)/(4))^2 + (y + (5)/(4))^2 = (4+9+25)/(16) (x - (3)/(4))^2 + (y + (5)/(4))^2 = (38)/(16) (x - (3)/(4))^2 + (y + (5)/(4))^2 = (19)/(8) Step 5: Identify the center and radius from the standard form. Comparing (x - (3)/(4))^2 + (y + (5)/(4))^2 = (19)/(8) with (x-h)^2 + (y-k)^2 = r^2: The center (h,k) is ((3)/(4), -(5)/(4)). The radius squared r^2 is (19)/(8). The radius r is sqrt((19)/(8)). To simplify the radius, rationalize the denominator: r = sqrt((19)/(8)) = sqrt(19)sqrt(8) = sqrt(19)2sqrt(2) = sqrt(19) · sqrt(2)2sqrt(2) · sqrt(2) = sqrt(38)2 · 2 = sqrt(38)4 The center of the circle is ((3)/(4), -(5)/(4)) and the length of the radius is sqrt(38)4. Send me the next one 📸