Question

question 18
identify a, b, h, and k for the ellipse graphed above and with an equation of the form:
\\( \frac { ( x - h ) ^ { 2 } } { a ^ { 2 } } + \frac { ( y - k ) ^ { 2 } } { b ^ { 2 } } = 1 \\)
\\( a = \\)
\\( b = \\)
\\( h = \\)
\\( k = \\)
question help: video 1 video 2

Explanation:

Step1: Find the center \((h,k)\)

The center of the ellipse is the mid - point. From the graph, the center is at \((3, - 1)\), so \(h = 3\) and \(k=-1\).

Step2: Find \(a\)

\(a\) is the distance from the center to the right (or left) vertex. The right vertex is at \((10,-1)\) and the center is at \((3,-1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), since \(y_1 = y_2=-1\), \(a=\vert10 - 3\vert=7\).

Step3: Find \(b\)

\(b\) is the distance from the center to the upper (or lower) vertex. The upper vertex is at \((3,7)\) and the center is at \((3,-1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), since \(x_1 = x_2 = 3\), \(b=\vert7-(-1)\vert=8\).

Answer:

\(a = 7\), \(b = 8\), \(h = 3\), \(k=-1\)