Question

match the graphs to their equations.
\frac{(x - 3)^{2}}{4}+\frac{(y - 1)^{2}}{9}=1
\frac{(x + 3)^{2}}{4}+\frac{(y + 1)^{2}}{9}=1
\frac{(x + 3)^{2}}{9}+\frac{(y - 1)^{2}}{4}=1
\frac{(x - 3)^{2}}{9}+\frac{(y + 1)^{2}}{4}=1

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}} = 1\). If \(a>b\), the major axis is horizontal, and if \(b > a\), the major axis is vertical. The center of the ellipse is \((h,k)\).

Step2: Analyze the first equation \(\frac{(x - 3)^{2}}{4}+\frac{(y - 1)^{2}}{9}=1\)

Here, \(h = 3\), \(k = 1\), \(a^{2}=4\) (\(a = 2\)), \(b^{2}=9\) (\(b=3\)). Since \(b>a\), the major axis is vertical. The center is \((3,1)\).

Step3: Analyze the second equation \(\frac{(x + 3)^{2}}{4}+\frac{(y + 1)^{2}}{9}=1\)

Here, \(h=-3\), \(k = - 1\), \(a^{2}=4\) (\(a = 2\)), \(b^{2}=9\) (\(b = 3\)). Since \(b>a\), the major axis is vertical. The center is \((-3,-1)\).

Step4: Analyze the third equation \(\frac{(x + 3)^{2}}{9}+\frac{(y - 1)^{2}}{4}=1\)

Here, \(h=-3\), \(k = 1\), \(a^{2}=9\) (\(a = 3\)), \(b^{2}=4\) (\(b = 2\)). Since \(a>b\), the major axis is horizontal. The center is \((-3,1)\).

Step5: Analyze the fourth equation \(\frac{(x - 3)^{2}}{9}+\frac{(y + 1)^{2}}{4}=1\)

Here, \(h = 3\), \(k=-1\), \(a^{2}=9\) (\(a = 3\)), \(b^{2}=4\) (\(b = 2\)). Since \(a>b\), the major axis is horizontal. The center is \((3,-1)\).

Step6: Match the centers and axes to the graphs

• For the equation \(\frac{(x - 3)^{2}}{9}+\frac{(y + 1)^{2}}{4}=1\), the center is \((3,-1)\) and major axis is horizontal. This matches graph a.
• For the equation \(\frac{(x - 3)^{2}}{4}+\frac{(y - 1)^{2}}{9}=1\), the center is \((3,1)\) and major axis is vertical. This matches graph b.
• For the equation \(\frac{(x + 3)^{2}}{9}+\frac{(y - 1)^{2}}{4}=1\), the center is \((-3,1)\) and major axis is horizontal. This matches graph c.
• For the equation \(\frac{(x + 3)^{2}}{4}+\frac{(y + 1)^{2}}{9}=1\), the center is \((-3,-1)\) and major axis is vertical. This matches graph d.

Answer:

\(\frac{(x - 3)^{2}}{9}+\frac{(y + 1)^{2}}{4}=1\) matches graph a; \(\frac{(x - 3)^{2}}{4}+\frac{(y - 1)^{2}}{9}=1\) matches graph b; \(\frac{(x + 3)^{2}}{9}+\frac{(y - 1)^{2}}{4}=1\) matches graph c; \(\frac{(x + 3)^{2}}{4}+\frac{(y + 1)^{2}}{9}=1\) matches graph d.