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The equation of an ellipse is given below. ((x-12)^(2))/(289)+((y-3)^(2))/(64)=1 What are the foci of this ellipse? Choose 1 answer: (A) (27,3) and (-3,3) (B) (3,27) and (3,-3) (c) (12,18) and (12,-12) (D) (18,12) and (-12,12)

The equation of an ellipse is given below.\newline(x−12)2289+(y−3)264=1 \frac{(x-12)^{2}}{289}+\frac{(y-3)^{2}}{64}=1 \newlineWhat are the foci of this ellipse?\newlineChoose 11 answer:\newline(A) (27,3) (27,3) and (−3,3) (-3,3) \newline(B) (3,27) (3,27) and (3,−3) (3,-3) \newline(C) (12,18) (12,18) and (12,−12) (12,-12) \newline(D) (18,12) (18,12) and (−12,12) (-12,12)

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Q. The equation of an ellipse is given below.\newline(x−12)2289+(y−3)264=1 \frac{(x-12)^{2}}{289}+\frac{(y-3)^{2}}{64}=1 \newlineWhat are the foci of this ellipse?\newlineChoose 11 answer:\newline(A) (27,3) (27,3) and (−3,3) (-3,3) \newline(B) (3,27) (3,27) and (3,−3) (3,-3) \newline(C) (12,18) (12,18) and (12,−12) (12,-12) \newline(D) (18,12) (18,12) and (−12,12) (-12,12)
  1. Identify major and minor axes: The given equation of the ellipse is (x−12)2289+(y−3)264=1\frac{(x-12)^{2}}{289}+\frac{(y-3)^{2}}{64}=1. To find the foci, we first need to identify the major and minor axes of the ellipse.
  2. Standard form of the ellipse equation: The standard form of the ellipse equation is (x−h)2/a2+(y−k)2/b2=1(x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k)(h, k) is the center of the ellipse, aa is the semi-major axis, and bb is the semi-minor axis. In the given equation, h=12h=12, k=3k=3, a2=289a^2=289, and b2=64b^2=64.
  3. Calculate semi-major and semi-minor axes: Since a2=289a^2=289, we find that a=289=17a=\sqrt{289}=17. Similarly, since b2=64b^2=64, we find that b=64=8b=\sqrt{64}=8. We can see that a > b, which means that the major axis is along the x-direction.
  4. Determine major axis direction: The foci of an ellipse are located along the major axis at a distance of cc from the center, where cc is found using the equation c2=a2−b2c^2 = a^2 - b^2. Let's calculate cc.
  5. Calculate distance of foci from center: Substitute the values of aa and bb into the equation c2=a2−b2c^2 = a^2 - b^2 to find cc.c2=172−82c^2 = 17^2 - 8^2c2=289−64c^2 = 289 - 64c2=225c^2 = 225c=225c = \sqrt{225}c=15c = 15
  6. Substitute values to find cc: The foci are located at a distance of cc from the center along the x-axis. Since the center is at (12,3)(12, 3), the foci will be at (12±c,3)(12\pm c, 3). Substituting the value of cc, we get the coordinates of the foci as (12±15,3)(12\pm 15, 3).
  7. Find coordinates of foci: Calculate the actual coordinates of the foci by adding and subtracting cc from the xx-coordinate of the center.\newlineFirst focus: (12+15,3)=(27,3)(12+15, 3) = (27, 3)\newlineSecond focus: (12−15,3)=(−3,3)(12-15, 3) = (-3, 3)

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