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

Answer 1

We have to write it in the standard form:

#4x^2-25y^2-50y-125=0rArr#
#4x^2-25(y^2+2y)=125rArr#
#4x^2-25(y^2+2y+1-1)=125rArr#
#4x^2-25(y+1)^2+25=125rArr#
#4x^2-25(y+1)^2=100rArr#
#4/100x^2-25/100(y+1)^2=1rArr#
#x^2/25-(y+1)^2/4=1#
This is an hyperbola centered in #C(0,-1)#, with the branches left and right (there is an #1# at the second member), with the asymptotes:
#m=+-4/25#, so:
#y-(-1)=+-4/25(x-0)rArry=+-4/25x-1#.

graph{4x^2-25y^2-50y-125=0 [-10, 10, -5, 5]}

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Answer 2

To graph the hyperbola with the equation (4x^2 - 25y^2 - 50y - 125 = 0), you'll first need to rewrite the equation in standard form, which is typically in the form (\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1) or (\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1), where ((h,k)) is the center of the hyperbola.

To accomplish this, complete the square for both the (x) and (y) terms separately. Once you've rewritten the equation in standard form, you can identify the center, vertices, foci, asymptotes, and sketch the hyperbola accordingly.

Remember, for a hyperbola with equation (\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1):

  • The center is at ((h,k)).
  • The vertices are ((h \pm a, k)).
  • The foci are ((h \pm c, k)), where (c = \sqrt{a^2 + b^2}).
  • The slopes of the asymptotes are (\pm \frac{b}{a}).

Follow these steps to graph the hyperbola:

  1. Rewrite the equation in standard form.
  2. Identify the values of (h), (k), (a), and (b).
  3. Find the center, vertices, foci, and slopes of asymptotes.
  4. Sketch the hyperbola using this information.

Once you've completed these steps, you'll have a graph of the hyperbola described by the given equation.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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