Elements of a hyperbola

Statement

Find the foci, vertices and asymptotes of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1.$

Solution

From the canonical equation one reads $a^2=9$ and $b^2=16$, so $a=3$ and $b=4$. The focal parameter is $c=\sqrt{a^2+b^2}=\sqrt{9+16}=\sqrt{25}=5.$ Therefore:

• Vertices: $V_{1,2}(\mp 3;0)$, that is $\boxed{V_1(-3;0),\ V_2(3;0)}$;
• Foci: $F_{1,2}(\mp 5;0)$, that is $\boxed{F_1(-5;0),\ F_2(5;0)}$;
• Asymptotes: $y=\pm\tfrac{b}{a}x=\boxed{\pm\tfrac{4}{3}x}$.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Foci · Hyperbola · Vertices
Functions: Hyperbola
Methods: Canonical hyperbola
Skills: Analytic geometry
Exercise type: Geometric problem