Identify the graph of 4x^2+5y^2=20 for T(5,-6) and write an equation of the translated or rotated graph in general form..

Answer:The answer is ellipse of equation 4x² + 5y² - 40x + 60y + 260 = 0 ⇒ answer (b)Step-by-step explanation:* At first lets talk about the general form of the conic equation- Ax² + Bxy + Cy²  + Dx + Ey + F = 0∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse. ∵ B² - 4AC = 0 , if a conic exists, it will be a parabola. ∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.* Now we will study our equation:* 4x² + 5y² = 20∵ A = 4 , B = 0 , C = 5∴ B² - 4 AC = (0)² - 4(4)(5) = -80∴ B² - 4AC < 0∴ The graph is ellipse or circle* If A and C are nonzero, have the same sign, and are not  equal to each other, then the graph is an ellipse.* If A and C are equal and nonzero and have the same  sign, then the graph is a circle.∵ A and C have same signs with different values∴ It is an ellipse* Now lets study T(5 , -6), that means the graph will translate  5 units to the right and 6 units down∴ x will be (x - 5) and y will be (y - -6) = (y + 6)* Lets substitute the x by ( x - 5) and y by (y + 6) in the equation∴ 4(x - 5)² + 5(y + 6)² = 20 * Use the foil method∴ 4(x² - 10x + 25) + 5(y² + 12y + 36) = 20* Open the brackets∴ 4x² - 40x + 100 + 5y² + 60y + 180 = 20* Collect the like terms∴ 4x² + 5y² - 40x + 60y + 280 = 20∴ 4x² + 5y² - 40x + 60y + 280 - 20 = 0∴ 4x² + 5y² - 40x + 60y + 260 = 0* The answer is ellipse of equation 4x² + 5y² - 40x + 60y + 260 = 0