Problem 36 - Algebra and Trigonometry

Question

Find an equation for the ellipse that shares a vertex and a focus with the parabola \(x^{2}+y=100\) and has its other focus at the origin.

Step-by-Step Solution

Answer

The ellipse shares a focus with the parabola at \( (0, -99.75) \) and lies along the y-axis, becoming degenerate.

Step 1: Identify the vertex of the parabola

The given parabola is described by the equation \(x^2 + y = 100\). Rewriting this in vertex form gives \(y = x^2 - 100\). The parabola opens upwards and its vertex, \(V\), is at \((0, -100)\).

Step 2: Find the focus of the parabola

Since the parabola is in the form \(y = x^2 - 100\), and it can be rewritten as \((x-0)^2 = 1(y + 100)\), the standard form \((x-h)^2 = 4p(y-k)\) tells us \(4p = 1\), so \(p = \frac{1}{4}\). The focus \(F_p\) of a parabola \(( (h, k + p) )\) is at \((0, -99.75)\).

Step 3: Identify shared elements and additional focus

The ellipse shares a vertex with the parabola, so one vertex is \((0, -100)\). It also shares a focus with the parabola at \((0, -99.75)\). The other focus of the ellipse is at the origin \((0,0)\).

Step 4: Determine the center of the ellipse

The centre of the ellipse, \(C\), is the midpoint of the two foci \((0, -99.75)\) and \((0, 0)\). Thus, \(C\) is located at \((0, -49.875)\).

Step 5: Determine the semi-major axis length

The rectangular distance between the foci is found from \((0, -99.75)\) to \((0, 0)\), which is 99.75. Hence, the semi-major axis length, \(a\), is half that, giving \(a = 49.875\).

Step 6: Calculate the semi-minor axis length

Using the ellipse property \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center to a focus, we have \(c = 49.875\). Since both foci are on the y-axis, \(b = 0\).

Step 7: Write the equation of the ellipse

The standard form of an ellipse’s equation with center \((h, k)\) and vertically-aligned axes is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\). Plug in the values: \(h = 0\), \(k = -49.875\), \(a = 49.875\), \(b = 0\). The final form is \(\frac{x^2}{0} + \frac{(y+49.875)^2}{2487.28} = 1\). This simplifies to \(x^2 = 0\), making it degenerate in this specific direction.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a point where the curve changes direction and is located at the good midpoint of the parabola. For the given parabola described by the equation \(x^2 + y = 100\), rewriting it in the vertex form as \(y = x^2 - 100\) helps us identify the vertex easily.

The vertex reflects the point \((h, k)\) where \(h\) is the x-coordinate, and \(k\) is the y-coordinate of the vertex. In this equation, the vertex is at \((0, -100)\), meaning the parabola opens upwards with the lowest point at \((0, -100)\).

• Vertex form of a parabola is \((x-h)^2 = 4p(y-k)\)
• The vertex is \( (h, k) \)
• Given equation: \(y = x^2 - 100\), vertex \(V = (0, -100)\)

Focus of a Parabola

The focus of a parabola is a critical component that, along with the directrix, defines precisely how the parabola curves. For the parabola \(y = x^2 - 100\), the focus can be identified by comparing it with the standard form \((x-h)^2 = 4p(y-k)\). Here, \(4p = 1\), so that means \(p = \frac{1}{4}\).

Knowing \(p\), where \(p\) is the distance from the vertex to the focus, we can locate the focus \(F_p\). The focus is \((h, k + p)\), which calculates to \((0, -99.75)\) for this parabola.

• Focus is located a distance \(p\) away from the vertex along the axis of symmetry
• Important in determining the 'sharpness' of the parabola
• Given focus: \((0, -99.75)\)

Semi-Major and Semi-Minor Axes

Ellipses are defined by their axes – the semi-major axis and semi-minor axis – which determine the shape and size of the ellipse. The semi-major axis is the longest radius of the ellipse, extending from its center to the furthest point on the perimeter, while the semi-minor is perpendicular and shorter.

For the ellipse with a shared focus at the origin and a second focus at \((0, -99.75)\), the center is at the midpoint of these foci, positioned at \((0, -49.875)\). This gives us the distance between the foci, \(c\), which is \(99.75\). The semi-major axis, \(a\), is half this distance: \(49.875\).

• Semi-Major Axis: Longest radius of the ellipse, \(a = 49.875\)
• Semi-Minor Axis: Typically perpendicular to the semi-major, but here the ellipse is stretched along the y-axis
• Focus relationship holds: \(c^2 = a^2 - b^2\)

Since this ellipse is defined vertically with these particular characteristics, its semi-minor axis is effectively collapsed to zero in this specific case, leading to a degenerate form.