Question

Find the distance from the vertex of the parabola \(g(x)=-3 x^{2}+6 x+1\) to the center of the circle \(x^{2}+y^{2}+10 x+8 y+32=0\)

Short answer

The distance is 10.

Step-by-step solution

Identify the vertex of the parabola

To find the vertex of the parabola given by the equation \(g(x) = -3x^2 + 6x + 1\), use the vertex formula \(x = -\frac{b}{2a}\). In this case, \(a = -3\) and \(b = 6\). Plug these values into the formula to get \[ x = -\frac{6}{2(-3)} = 1 \]. Now, substitute \(x = 1\) back into the equation to find the y-coordinate of the vertex: \[ y = -3(1)^2 + 6(1) + 1 = -3 + 6 + 1 = 4 \]. Therefore, the vertex of the parabola is \((1, 4)\).

Identify the center of the circle

The equation of the circle is given by \(x^2 + y^2 + 10x + 8y + 32 = 0\). To find the center, rewrite it in standard form by completing the square. First, group the \(x\) and \(y\) terms: \((x^2 + 10x) + (y^2 + 8y) = -32\). Complete the square for both groups:\(x^2 + 10x\) becomes \((x + 5)^2 - 25\) and \(y^2 + 8y\) becomes \((y + 4)^2 - 16\). Substitute these back into the equation:\((x + 5)^2 - 25 + (y + 4)^2 - 16 = -32\) simplifies to \((x + 5)^2 + (y + 4)^2 = 9\). The center of the circle is \((-5, -4)\).

Calculate the distance between the vertex and the center

Use the distance formula to find the distance between the vertex of the parabola \((1, 4)\) and the center of the circle \((-5, -4)\): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Substituting \(x_1 = 1\), \(y_1 = 4\), \(x_2 = -5\), and \(y_2 = -4\): \[ d = \sqrt{(-5 - 1)^2 + (-4 - 4)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]. Therefore, the distance is 10.

Key concepts

vertex of parabola

The vertex of a parabola is a crucial point. It represents the peak of the graph if the parabola opens down, and the lowest point if it opens up. The vertex can be found using the vertex formula for a given quadratic equation in the form \(ax^2 + bx + c\). The formula is \(x = -\frac{b}{2a}\).

For our quadratic equation \(g(x) = -3x^2 + 6x + 1\), \(a = -3\) and \(b = 6\). Plugging these values into the vertex formula, we get \(x = -\frac{6}{2(-3)} = 1\).

Once we have the \(x\)-coordinate, substitute it back into the original equation to get the \(y\)-coordinate. Thus, \(y = -3(1)^2 + 6(1) + 1 = 4\). Therefore, the vertex of the parabola is \((1, 4)\).

center of circle

To find the center of a circle given by a general form equation, we need to rewrite it in standard form through a technique called completing the square. The general form of the circle's equation is \(x^2 + y^2 + 10x + 8y + 32 = 0\).

First group the \(x\) and \(y\) terms together: \((x^2 + 10x) + (y^2 + 8y) = -32\). Now, complete the square for each group. \(x^2 + 10x\) becomes \((x + 5)^2 - 25\) and \(y^2 + 8y\) becomes \((y + 4)^2 - 16\).

Substitute these back into the circle equation: \((x + 5)^2 - 25 + (y + 4)^2 - 16 = -32\). Simplifying, we get \((x + 5)^2 + (y + 4)^2 = 9\), indicating the center of the circle is \((-5, -4)\).

distance formula

The distance formula calculates the distance between two points in a coordinate plane. It is derived from the Pythagorean Theorem and is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

For our problem, we need to find the distance between the vertex of the parabola \((1, 4)\) and the center of the circle \((-5, -4)\). Substituting \(x_1 = 1\), \(y_1 = 4\), \(x_2 = -5\), and \(y_2 = -4\) into the formula, we have: \(d = \sqrt{(-5 - 1)^2 + (-4 - 4)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\).

Therefore, the distance between the vertex of the parabola and the center of the circle is \(10\) units.

completing the square

Completing the square is a method used to convert a quadratic expression into a perfect square trinomial. This technique makes it easier to identify key features of graphs and solve quadratic equations.

For the circle’s equation given as \(x^2 + y^2 + 10x + 8y + 32 = 0\), we first separate the \(x\) and \(y\) terms: \((x^2 + 10x) + (y^2 + 8y) = -32\).

Next, we complete the square for each grouped terms:

• For \(x^2 + 10x\), add and subtract \(25\) (i.e., \(10/2 = 5\) and \(5^2 = 25\)), giving us \((x + 5)^2 - 25\).
• For \(y^2 + 8y\), add and subtract \(16\) (i.e., \(8/2 = 4\) and \(4^2 = 16\)), giving us \((y + 4)^2 - 16\).

Now, substitute back and simplify: \((x + 5)^2 - 25 + (y + 4)^2 - 16 = -32\). This simplifies further to: \((x + 5)^2 + (y + 4)^2 = 9\).

The equation now clearly shows the circle’s center is \((-5, -4)\) and its radius is \(3\) (as the square root of \(9\) is \(3\)).