If `a, h`, and `k` are nonzero constants, and the parabola with equation `y=a(x-h)^(2)+k`, in the xy-plane, passes through the point `(h, 5)` and `(0, 2)` , which of the following must be true ?

Options

A. `h^(2)= - 3/a`
B. `a^(2)= - 3/h`
C. `a= - 3/h`
D. `ah= -3`

Answer

To solve the problem, we need to determine the values of the constants \(a\), \(h\), and \(k\) based on the given points that the parabola passes through: \((h, 5)\) and \((0, 2)\). We will use the equation of the parabola, which is given by:

\[ y = a(x - h)^2 + k \]

### Step 1: Substitute the first point \((h, 5)\)

Since the point \((h, 5)\) lies on the parabola, we substitute \(x = h\) and \(y = 5\) into the equation:

\[ 5 = a(h - h)^2 + k \]

This simplifies to:

\[ 5 = a(0)^2 + k \]

Thus, we have:

\[ 5 = k \quad \text{(Equation 1)} \]

### Step 2: Substitute the second point \((0, 2)\)

Next, we substitute the second point \((0, 2)\) into the equation of the parabola. Here, we set \(x = 0\) and \(y = 2\):

\[ 2 = a(0 - h)^2 + k \]

This simplifies to:

\[ 2 = a(-h)^2 + k \]

Since \((-h)^2 = h^2\), we can rewrite it as:

\[ 2 = ah^2 + k \]

### Step 3: Substitute \(k\) from Equation 1

Now, we substitute \(k = 5\) from Equation 1 into the equation we derived from the second point:

\[ 2 = ah^2 + 5 \]

### Step 4: Solve for \(a\)

Now, we can isolate \(a\):

\[ 2 - 5 = ah^2 \]

This simplifies to:

\[ -3 = ah^2 \]

Rearranging gives us:

\[ a = \frac{-3}{h^2} \quad \text{(Equation 2)} \]

### Conclusion

From our calculations, we have determined that \(k = 5\) and \(a = \frac{-3}{h^2}\).

### Final Result

The relationships we found must hold true for the constants \(a\), \(h\), and \(k\) given the points through which the parabola passes.