This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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a) Write the ellipse equation in standard form. Step 1: Identify the center and value of c. The foci are F_1(2,0) and F_2(-2,0). The center of the ellipse is the midpoint of the foci: Center = ((2 + (-2))/(2), (0 + 0)/(2)) = (0,0) The distance from the center to each focus is c. c = 2 Since the foci are on the x-axis, the major axis is horizontal. The standard form of the ellipse is (x^2)/(a^2) + (y^2)/(b^2) = 1. Step 2: Use the latus rectum formula. The latus rectum L = 6. The formula for the latus rectum is L = (2b^2)/(a). 6 = (2b^2)/(a) 3a = b^2 (1) Step 3: Use the relationship between a, b, c. For an ellipse, c^2 = a^2 - b^2. Substitute c=2: 2^2 = a^2 - b^2 4 = a^2 - b^2 (2) Step 4: Solve the system of equations for a^2 and b^2. Substitute equation (1) into equation (2): 4 = a^2 - 3a Rearrange into a quadratic equation: a^2 - 3a - 4 = 0 Factor the quadratic equation: (a-4)(a+1) = 0 Since a must be positive, a=4. Now find b^2 using equation (1): b^2 = 3a = 3(4) = 12 And a^2 = 4^2 = 16. Step 5: Write the equation of the ellipse. Substitute a^2 = 16 and b^2 = 12 into the standard form: (x^2)/(16) + (y^2)/(12) = 1 The equation of the ellipse in standard form is (x^2)/(16) + (y^2)/(12) = 1. b) Find the centre and radius of the circle: 36x^2 + 36y^2 - 24x - 36y - 23 = 0 using completing of the square method. Step 1: Divide the entire equation by 36 to make the coefficients of x^2 and y^2 equal to 1. (36x^2)/(36) + (36y^2)/(36) - (24x)/(36) - (36y)/(36) - (23)/(36) = 0 x^2 + y^2 - (2)/(3)x - y - (23)/(36) = 0 Step 2: Group the x terms and y terms, and move the constant term to the right side. (x^2 - (2)/(3)x) + (y^2 - y) = (23)/(36) Step 3: Complete the square for the x terms and y terms. For the x terms: Take half of the coefficient of x (-(2)/(3)), which is -(1)/(3), and square it: (-(1)/(3))^2 = (1)/(9). For the y terms: Take half of the coefficient of y (-1), which is -(1)/(2), and square it: (-(1)/(2))^2 = (1)/(4). Step 4: Add these values to both sides of the equation. (x^2 - (2)/(3)x + (1)/(9)) + (y^2 - y + (1)/(4)) = (23)/(36) + (1)/(9) + (1)/(4) Step 5: Rewrite the left side as squared terms and simplify the right side. (x - (1)/(3))^2 + (y - (1)/(2))^2 = (23)/(36) + (4)/(36) + (9)/(36) (x - (1)/(3))^2 + (y - (1)/(2))^2 = (23 + 4 + 9)/(36) (x - (1)/(3))^2 + (y - (1)/(2))^2 = (36)/(36) (x - (1)/(3))^2 + (y - (1)/(2))^2 = 1 Step 6: Identify the center and radius from the standard form of a circle (x-h)^2 + (y-k)^2 = r^2. The center (h,k) is ((1)/(3), (1)/(2)). The radius squared r^2 = 1, so the radius r = sqrt(1) = 1. The center of the circle is ((1)/(3), (1)/(2)) and the radius is 1.