A hyperbola is centered at the origin with transverse axis along the -axis. Its real axis length is and its eccentricity is . Find the equations of its asymptotes.

Analytic Geometry Solution – Problem 31: Find the equations of its asymptotes

Question

A hyperbola is centered at the origin with transverse axis along the $x$ -axis. Its real axis length is $4$ and its eccentricity is $e=\dfrac54$ .

Find the equations of its asymptotes.

Step-by-step solution

(1) The real axis length is $2a=4\Rightarrow a=2$ .

(2) Eccentricity $e=\dfrac{c}{a}=\dfrac54\Rightarrow c=ae=2\cdot\frac54=\frac52$ .

(3) For a hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ , we have $c^2=a^2+b^2$ . Thus $b^2=c^2-a^2=\left(\frac52\right)^2-2^2=\frac{25}{4}-4=\frac94,\quad b=\frac32.$ (4) The asymptotes are $y=\pm\frac{b}{a}x=\pm\frac{3/2}{2}x=\pm\frac34x.$

Final answer

The asymptotes are $y=\pm\dfrac34x$ .

Marking scheme

Step 1 — Setup

Checkpoint: identify $a=2$ from the real axis length and set $c=ae$ (2 pts)

Step 2 — Key Calculation

Checkpoint: compute $b^2=c^2-a^2=\frac94$ and use asymptote slope $\pm\frac{b}{a}$ (3 pts)

Step 3 — Final Answer

Checkpoint: state $y=\pm\frac34x$ (2 pts)

Zero credit if: uses ellipse relation $c^2=a^2-b^2$ for a hyperbola.

Deductions: -1 pt for arithmetic error in $b$ .

Analytic Geometry – Problem 31: Find the equations of its asymptotes