Question

Find the focus and directrix of the parabola $$ x^{2}-6 x+4 y+3=0 $$

Short answer

Focus: \((3, 0.5)\), Directrix: \(y = 2.5\)

Step-by-step solution

Rearrange the Equation

Start by rearranging the given equation of the parabola, \(x^2 - 6x + 4y + 3 = 0\), to separate the \(x\) terms from the \(y\) terms. Move \(3\) to the other side: \[ x^2 - 6x + 4y = -3 \]

Complete the Square

To handle the quadratic in \(x\), complete the square. The expression is \(x^2 - 6x\). 1. Take half of the coefficient of \(x\), which is \(-6\), half of \(-6\) is \(-3\), and square it to get \(9\).2. Add and subtract \(9\) inside the equation: \( x^2 - 6x + 9 - 9 + 4y = -3 \).3. Group the perfect square trinomial and simplify: \((x - 3)^2 - 9 + 4y = -3\) \[(x - 3)^2 + 4y = 6\]

Isolate the Parabola Block

Make the equation resemble the standard form \((x-h)^2 = 4p(y-k)\). Rearrange the equation \((x - 3)^2 = -4(y - (6/4))\) to show \((x - 3)^2 = -4(y - 1.5)\).

Identify Parameters of Parabola

From the equation \((x - 3)^2 = -4(y - 1.5)\), identify:- The vertex \((h, k) = (3, 1.5)\).- Since \(4p = -4\), solve for \(p\): \(p = -1\).

Determine the Focus

The focus of a parabola \((x-h)^2 = 4p(y-k)\) is at the point \((h, k+p)\). Here, it is \((3, 1.5 - 1) = (3, 0.5)\).

Determine the Directrix

The directrix is the line \(y = k-p\). Here, it is \(y = 1.5 + 1\) or \(y = 2.5\).

Key concepts

Focus

The focus in the context of a parabola is a crucial point that is used to define the curve's geometric properties. A parabola is formed by the collection of all points that are equidistant from a single point (the focus) and a line (the directrix). This unique point influences the shape and position of the parabola.
In our example, the parabola is given by the equation in its standard form \( (x - 3)^2 = -4(y - 1.5) \). The focus can be found using the coordinates of the vertex \((h, k)\) and the parameter \(p\). Here, the vertex is \((3, 1.5)\). The value of \(p\) is \(-1\). So the focus, located \(p\) units away from the vertex along the axis of symmetry, is \((3, 0.5)\).
Remember that the focus always lies along the direction the parabola "opens," which in this case means it is below the vertex since \(p\) is negative.

Directrix

The directrix is a significant straight line related to the parabola. It is the reference line such that every point on the parabola is equidistant from it and the focus. Understanding the directrix helps determine how "wide" or "narrow" the parabola is.
For our parabola with vertex \((3, 1.5)\) and \(p = -1\), the directrix is calculated by the formula for a vertically-oriented parabola: \(y = k - p\). By substituting the values, we find \(y = 1.5 + 1 = 2.5\). Hence, the directrix is the horizontal line \(y = 2.5\), which is located above the vertex due to the negative \(p\) value.
This line helps define the overall boundary of the parabola in relation to its focal point.

Completing the Square

Completing the square is a useful algebraic technique used to transform a quadratic equation into a format that reveals key properties of the graph, specifically the vertex form of a parabola. This method involves creating a perfect square trinomial to easily rewrite the quadratic expression.
In the original exercise, to complete the square for \(x^2 - 6x\):

• Take half of the linear coefficient, \(-6\), which gives \(-3\).
• Square \(-3\) to obtain 9.
• Add and subtract \(9\) to balance the equation:\((x^2 - 6x + 9 - 9 + 4y = -3)\).
• Factor the resulting perfect square trinomial to get \((x - 3)^2\).

Thus, we simplify the expression to \((x - 3)^2 + 4y = 6\).
This straightforward process is essential as it allows us to find the vertex and reorganize the equation into a more manageable form.

Vertex Form

The vertex form of a parabola provides an easy way to identify the vertex point and the direction(s) the parabola opens. It's written as \((x - h)^2 = 4p(y - k)\) for vertical parabolas, where \((h, k)\) is the vertex.
For our parabola equation \((x - 3)^2 = -4(y - 1.5)\), this form was derived by completing the square. This transformation is useful since:

• The vertex is directly visible as \((h, k) = (3, 1.5)\).
• The value of \(4p\) tells us about the distance and direction between the focus and directrix. Here, \(4p = -4\), giving \(p = -1\), indicating a downward opening.

Understanding the vertex form is essential as it simplifies the process of graphing the parabola and analyzing its geometry.