Problem 90 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find the standard form of the equation of the hyperbola with vertices \((5,-6)\) and \((5,6),\) passing through \((0,9)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola whose points are equidistant from \(y=4\) and \((-1,0)\) is \(y = -\frac{1}{8}(x+1)² + 2\).

Step 1: Determine the Vertex

Given that points on the parabola are equidistant from \(y = 4\) and \((-1,0)\), the vertex of the parabola can be determined by finding the midpoint. The line \(y = 4\) is parallel to the x - axis, and \((-1,0)\) is a point in the plane. Additionally, the parabola opens upwards or downwards because it is equidistant from a point and a line. Therefore, the y - coordinate of the vertex is the midpoint of 0 and 4, which equals \(\frac{4+0}{2} = 2\). Since the line is parallel to the x - axis, the x - coordinate of the vertex is the same as the x - coordinate of the point, -1. Therefore, the vertex (h,k) is (-1,2).

Step 2: Determine the Direction of Parabola

The parabola opens up if the focus is above the directrix and opens down if the focus is below the directrix. In this scenario, since the directrix, \(y=4\), is above the focus, at \((-1,0)\), the parabola opens downwards.

Step 3: Determine the Value of 'a'

The absolute value of 'a' in the equation of a parabola equates to the reciprocal of 4 times the distance from the vertex to the directrix or the focus. Since the vertex of the parabola is equidistant from the focus and the directrix, the value of 'a' is \(\frac{1}{4d}\) where \(d = 2\) (distance from the vertex (2) to the directrix or focus (0 or 4)). Therefore, 'a' equals to \(\frac{1}{4*2} = \frac{1}{8}\). Because the parabola opens downwards, 'a' is negative. Thus, 'a' is -\frac{1}{8}.

Step 4: Write Down the Equation of Parabola

Now substitute the values of 'h', 'k', and 'a' into the standard parabolic equation. The equation is \(y = a(x-h)² + k = -\frac{1}{8}(x+1)² + 2\)

Key Concepts

Standard Form Equation

The standard form of a hyperbola's equation is crucial for understanding its geometric properties. In general, for a hyperbola centered at a point \(h, k\), its equation can be expressed as:

• Horizontally oriented hyperbola: \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]
• Vertically oriented hyperbola: \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]

Here, \(a\) and \(b\) are distances that influence the shape of the hyperbola. A hyperbola passing through a specific point, like \(0,9\), requires substituting these coordinates into the equation to solve for unknowns. Each standard form uniquely aligns the hyperbola with its axes, reflecting its symmetry and the direction of its opening.

Vertices

Vertices are vital points that define the extent of a hyperbola along its axis. They signal where the hyperbola changes direction. For a hyperbola centered at \(h, k\), the vertices have straightforward placements:

• Horizontally oriented: vertices at \(h\pm a, k\)
• Vertically oriented: vertices at \(h, k\pm a\)

The distance \(2a\) between the vertices marks a significant stretch of the hyperbola. Given vertices such as \(5,-6\) and \(5,6\), we can deduce the center as \(h = 5\), with a vertical orientation since the y-coordinates change while x-coordinates remain constant. Calculating \(a\) involves finding half the distance between the endpoints.

Conic Sections

Hyperbolas are a fascinating type of conic section. Conic sections arise when a plane intersects with a double cone. Depending on this intersection, one can produce circles, ellipses, parabolas, or hyperbolas.
Hyperbolas are unique because they consist of two open curves and form when the intersecting plane is parallel to the cone's axis. Unlike other conic sections, hyperbolas have two branches, formed because the intersection happens outside the cone's opposite side.

• Reflective properties of hyperbolas make them applicable in technology, like radio antennas.
• Recognizing how each part of the hyperbola relates to its equation aids in understanding its broader mathematical context.

Focus and Directrix

Understanding the focus and the directrix of a hyperbola deepens comprehension of its geometry.
The foci (focus points) are located along the transverse axis, with the formula \(c^2 = a^2 + b^2\) dictating their distance from the center. The directrices are lines used less in hyperbolas compared to other conics, yet they offer insight into how points on a hyperbola relate to its overall symmetry.

• The foci provide locations for maximum curvature, reinforcing the hyperbola's symmetric properties.
• By examining a point's distance from the focus and a directrix, one can verify its position on the hyperbola using the hyperbola's defining property.

Even though directrices are more fundamental in parabolas, recognizing their role alongside the foci helps in visualizing the hyperbola's structure and behavior.