Question

Show that the equation \(a \pm c x / b=\sqrt{x^{2}+y^{2}}\) is a parabola if \(b=c\), is an ellipse if \(b>c\), and is a hyperbola if \(b

Short answer

Answer: The given equation represents a parabola if b = c, an ellipse if b > c, and a hyperbola if b < c.

Step-by-step solution

Rewrite the given equation in terms of x and y

By moving the term \(\sqrt{x^2 + y^2}\) to the left-hand side of the equation, we obtain the following equation: \(\sqrt{x^2+y^2} - a \pm \frac{cx}{b}= 0\)

Square both sides

To eliminate the square root, square both sides of the equation: \((\sqrt{x^2+y^2} - a\pm \frac{cx}{b})^2 = (x^2+y^2) +a^2 \pm 2a\frac{cx}{b}+\frac{c^2x^2}{b^2} = x^2+y^2\)

Cancel terms and rearrange

The term \(x^2\) on both sides can be canceled. Then, solve the equation for \(y^2\) as follows, \(y^2 = a^2\pm2a\frac{cx}{b} + \frac{c^2x^2}{b^2}\)

Analyze the equation for each case

Now we will analyze the equation with the three given conditions. 1. If \(b = c\), we have: \(y^2 = a^2\pm2ax + x^2\) This is the equation for a parabola, with the squared variable (\(y^2\)) on one side and the non-squared variables on the other side. 2. If \(b > c\), consider w.l.o.g. \(b = kc\), where \(k > 1\). Then our equation takes the form: \(y^2 = a^2 \pm 2a\frac{cx}{kc} +\frac{c^2x^2}{k^2c^2}\) Which simplifies to: \(y^2 = a^2\pm 2a\frac{x}{k}+\frac{x^2}{k^2}\) Divide both sides by \(a^2\): \(\frac{y^2}{a^2} = 1\pm 2\frac{x}{ka} + \frac{x^2}{k^2a^2}\) By completing the square on the right-hand side, we obtain the equation of an ellipse: \(\frac{y^2}{a^2}+\frac{(x \mp \frac{a}{k})^2}{\frac{a^2}{k^2}}=1\) 3. If \(b < c\), consider w.l.o.g \(b = kc\), where \(k < 1\). Repeat the reasoning from case 2, but this time we obtain the equation of a hyperbola: $\frac{y^2}{a^2}-\frac{(x\mp \frac{a}{k})^2}{\frac{a^2}{k^2}}=1$ In conclusion, we have shown that the given equation represents a parabola if \(b = c\), an ellipse if \(b > c\), and a hyperbola if \(b < c\).

Key concepts

Parabola

A parabola is a U-shaped curve that you probably have seen in graphs of quadratic equations like \( y = ax^2 + bx + c \). Basically, a parabola is the set of all points that are equidistant from a fixed point, called the focus, and a straight line, called the directrix.
This concept makes parabolas quite unique since you only need one of these to describe several real-life phenomena like projectile motions.

For our specific equation, when \( b = c \), it simplifies to \( y^2 = a^2 \pm 2ax + x^2 \). This can be rearranged to follow the general structure of a parabola, indicating that the equation represents a parabolic shape. Unlike ellipses and hyperbolas, parabolas open indefinitely and don't loop back onto themselves or split in two. This open nature is exactly what defines the conic section known as a parabola.

Ellipse

An ellipse looks like a squished or elongated circle, and you’ve probably seen it when looking at the orbits of planets or the shape of a watermelon slice. In mathematical terms, an ellipse is defined as the set of all points for which the sum of the distances to two fixed points (called foci) is constant.

In our exercise, when the condition \( b > c \) is met, the equation transforms to the form \( \frac{y^2}{a^2} + \frac{(x \mp \frac{a}{k})^2}{\frac{a^2}{k^2}} = 1 \) after a series of rearrangements and simplifications. This is the standard equation of an ellipse. Unlike a parabola, an ellipse is a closed curve, which implies that the values are bounded and the trajectory loops back upon itself.

• Ellipses have two axes: the major axis (the longest diameter) and the minor axis (the shortest diameter).
• The foci lie along the major axis inside the ellipse.

The property that distinguishes ellipses from other conics is their symmetry and closed nature.

Hyperbola

A hyperbola may seem less familiar, but it's essentially two mirrored open curves pointing away from each other. Hyperbolas occur in real-world scenarios like the paths of comets near the sun or when navigating using GPS.

In mathematical terms, a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant.
When the equation condition changes to \( b < c \), the transformation yields the hyperbolic equation \( \frac{y^2}{a^2} - \frac{(x \mp \frac{a}{k})^2}{\frac{a^2}{k^2}} = 1 \). This form is typical of a hyperbola, where the presence of a negative sign distinguishes it from the formula for an ellipse.

Key characteristics of hyperbolas are:

• They consist of two separate branches.
• Each branch is a mirror image of the other.

Due to their shape, hyperbolas have asymptotes, which are lines they approach but never actually meet. This is quite different from the closed nature of ellipses and the simple open curve of parabolas.