Question

define hyperbola

provide an example:

Explanation:

Define Hyperbola:

A hyperbola is a type of conic section. In the plane, a hyperbola can be defined as the set of all points \((x,y)\) such that the absolute difference of the distances from \((x,y)\) to two distinct fixed points (called foci) is a constant. Mathematically, if the foci are \(F_1\) and \(F_2\), and a point on the hyperbola is \(P\), then \(| |PF_1| - |PF_2| | = 2a\), where \(2a\) is the constant difference and \(2c\) is the distance between the foci (\(c > a>0\)). The standard form of a hyperbola centered at the origin \((0,0)\) opening horizontally is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\), where \(c^{2}=a^{2}+b^{2}\), and for a hyperbola opening vertically, it is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\).

Provide an example:

Consider the hyperbola given by the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\). Here, \(a^{2} = 9\) (so \(a = 3\)) and \(b^{2}=16\) (so \(b = 4\)). We can find \(c\) using the relation \(c^{2}=a^{2}+b^{2}\), so \(c^{2}=9 + 16=25\) and \(c = 5\). The foci are at \((\pm c,0)=(\pm5,0)\). For any point \((x,y)\) on this hyperbola, the absolute difference of the distances from \((x,y)\) to \((5,0)\) and \((- 5,0)\) is \(2a=6\). For example, if we take the vertex \((3,0)\), the distance from \((3,0)\) to \((5,0)\) is \(2\) and the distance from \((3,0)\) to \((-5,0)\) is \(8\), and \(|8 - 2|=6 = 2a\). Another point, say when \(x = 5\), we can solve for \(y\): \(\frac{25}{9}-\frac{y^{2}}{16}=1\), \(\frac{y^{2}}{16}=\frac{25}{9}-1=\frac{16}{9}\), \(y^{2}=\frac{256}{9}\), \(y=\pm\frac{16}{3}\). The distance from \((5,\frac{16}{3})\) to \((5,0)\) is \(\frac{16}{3}\), and the distance from \((5,\frac{16}{3})\) to \((-5,0)\) is \(\sqrt{(5 + 5)^{2}+(\frac{16}{3}-0)^{2}}=\sqrt{100+\frac{256}{9}}=\sqrt{\frac{900 + 256}{9}}=\sqrt{\frac{1156}{9}}=\frac{34}{3}\). Then \(|\frac{34}{3}-\frac{16}{3}|=\frac{18}{3}=6 = 2a\), which satisfies the definition of a hyperbola.

Answer:

• Define Hyperbola: A hyperbola is the set of all points in a plane where the absolute difference of the distances from two fixed points (foci) is constant. Standard forms: \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\) (horizontal) or \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) (vertical), with \(c^{2}=a^{2}+b^{2}\) (\(c > a>0\)).
• Provide an example: The hyperbola \(\boldsymbol{\frac{x^{2}}{9}-\frac{y^{2}}{16}=1}\) (foci at \((\pm5,0)\), vertex at \((\pm3,0)\), and satisfies the distance - difference property).