How do I graph the hyperbola with the equation 4x^2−25y^2−50y−125=0?

Mar 2, 2015

We have to write it in the standard form:

$4 {x}^{2} - 25 {y}^{2} - 50 y - 125 = 0 \Rightarrow$

$4 {x}^{2} - 25 \left({y}^{2} + 2 y\right) = 125 \Rightarrow$

$4 {x}^{2} - 25 \left({y}^{2} + 2 y + 1 - 1\right) = 125 \Rightarrow$

$4 {x}^{2} - 25 {\left(y + 1\right)}^{2} + 25 = 125 \Rightarrow$

$4 {x}^{2} - 25 {\left(y + 1\right)}^{2} = 100 \Rightarrow$

$\frac{4}{100} {x}^{2} - \frac{25}{100} {\left(y + 1\right)}^{2} = 1 \Rightarrow$

${x}^{2} / 25 - {\left(y + 1\right)}^{2} / 4 = 1$

This is an hyperbola centered in $C \left(0 , - 1\right)$, with the branches left and right (there is an $1$ at the second member), with the asymptotes:

$m = \pm \frac{4}{25}$, so:

$y - \left(- 1\right) = \pm \frac{4}{25} \left(x - 0\right) \Rightarrow y = \pm \frac{4}{25} x - 1$.

graph{4x^2-25y^2-50y-125=0 [-10, 10, -5, 5]}
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}