Question

Find the distance from the vertex of the parabola \(f(x)=2(x-3)^{2}+5\) to the center of the circle \((x+3)^{2}+(y-1)^{2}=4\)

Short answer

The distance is \(2\sqrt{13}\).

Step-by-step solution

Identify the vertex of the parabola

The given parabola is in the vertex form: \(f(x) = 2(x-3)^2 + 5\). The vertex form of a parabola is \(f(x) = a(x-h)^2 + k\), where (h,k) is the vertex. Here, the vertex is at \((3, 5)\).

Identify the center of the circle

The given circle equation is \((x+3)^2 + (y-1)^2 = 4\). The standard form of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where (h, k) is the center. So, the center of the circle is at \((-3, 1)\).

Use the distance formula

To find the distance between the vertex \((3, 5)\) and the center of the circle \((-3, 1)\), use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the points \((x_1, y_1) = (3, 5)\) and \((x_2, y_2) = (-3, 1)\), we get: \[ d = \sqrt{(-3-3)^2+(1-5)^2} = \sqrt{(-6)^2+(-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

Key concepts

Vertex Form of a Parabola

The vertex form of a parabola is a useful way to describe the shape and position of a parabola. The general formula is \( f(x) = a(x-h)^2 + k \), where the point \((h, k)\) is the vertex of the parabola. This form makes it easy to see the key attributes:

Key Attributes

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.
• The vertex \((h,k)\) is the highest or lowest point on the graph.

Understanding the vertex form helps in quickly graphing the parabola and finding important points. In the given exercise, the parabola \( f(x) = 2(x-3)^2 + 5 \) shows us that the vertex is at \((3, 5)\). This information is crucial for solving any problem involving the parabola.

Standard Form of a Circle

The standard form of a circle's equation helps easily identify its center and radius. The general form is \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius. This makes it straightforward to graph the circle.

Key Attributes

• The point \((h, k)\) is the center of the circle.
• The radius is \( r \), which is the distance from the center to any point on the circle.

In our exercise, the circle equation \((x+3)^2+(y-1)^2=4\) tells us the center is at \((-3, 1)\) and the radius is \(2\). Knowing the center allows us to find distances from this point.

Distance Formula

The distance formula is a key tool in geometry to find the distance between two points in a plane. The formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Here's a step-by-step guide on how to use it:

Steps to Use the Distance Formula

• Identify the coordinates of the two points.
• Subtract the x-coordinates and y-coordinates: \( (x_2 - x_1) \) and \( (y_2 - y_1) \).
• Square the results: \( (x_2 - x_1)^2 \) and \( (y_2 - y_1)^2 \).
• Sum the squares: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \).
• Take the square root of the sum.

In the exercise, we needed to find the distance between the vertex \((3, 5)\) and the center of the circle \((-3, 1)\). Plugging these points into the formula gives us \[ d = \sqrt{(-3-3)^2+(1-5)^2} = \sqrt{(-6)^2+(-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]\.