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How to rewrite equation of hyperbola in standard form

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2

$begingroup$

I was wondering about this question:

$$ 9 x ^ 2 -4y^2-72x=0 $$

What is the step-by-step process of writing such an equation which, in this case, has the graph of a hyperbola in standard form?

Please excuse me for my messy equation. As I am relatively new to Mathematics Stack Exchange, I do not know how to insert superscripts.

Thank you ahead of time!

calculus conic-sections

share|cite|improve this question

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  In short: complete the square
  $endgroup$
  – Minus One-Twelfth
  2 hours ago
  
  
  

  
  
  
  

add a comment | 

2

$begingroup$

I was wondering about this question:

$$ 9 x ^ 2 -4y^2-72x=0 $$

What is the step-by-step process of writing such an equation which, in this case, has the graph of a hyperbola in standard form?

Please excuse me for my messy equation. As I am relatively new to Mathematics Stack Exchange, I do not know how to insert superscripts.

Thank you ahead of time!

calculus conic-sections

share|cite|improve this question

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  In short: complete the square
  $endgroup$
  – Minus One-Twelfth
  2 hours ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

I was wondering about this question:

$$ 9 x ^ 2 -4y^2-72x=0 $$

What is the step-by-step process of writing such an equation which, in this case, has the graph of a hyperbola in standard form?

Please excuse me for my messy equation. As I am relatively new to Mathematics Stack Exchange, I do not know how to insert superscripts.

Thank you ahead of time!

calculus conic-sections

share|cite|improve this question

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

$endgroup$

share|cite|improve this question

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

share|cite|improve this question

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

edited 2 hours ago

Key Flex

8,63761233

edited 2 hours ago

Key Flex

8,63761233

asked 2 hours ago

JamesJames

555

asked 2 hours ago

JamesJames

555

3 Answers
3

active

oldest

votes

4

$begingroup$

Note that $dfrac{(x-h)^2}{a^2}-dfrac{(y-k)^2}{b^2}=1$ is the standard form of hyperbola.

$$9x^2-4y^2-72x=0$$
$$9(x^2-8x)-4y^2=0$$
$$(x^2-8x)-dfrac49y^2=0$$
$$dfrac14(x^2-8x)-dfrac19y^2=0$$
$$dfrac14(x^2-8x+16)-dfrac19y^2=dfrac{1}{4}(16)$$
$$dfrac14(x-4)^2-dfrac19y^2=4$$
$$dfrac{(x-4)^2}{16}-dfrac{y^2}{36}=1$$
$$dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}=1mbox{ is the required Hyperbola}$$

share|cite|improve this answer

answered 2 hours ago

Key FlexKey Flex

8,63761233

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it not the equation before your answer that is in standard form since the 4^2 and 6^2 become 16 and 36. The equation with 16 & 36 as denominators.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @James $dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}$ is in the standard form.
  $endgroup$
  – Key Flex
  1 hour ago
  

  
  
  
  

add a comment | 

2

$begingroup$

So we have $$9(x^2-8x)-4y^2=0$$

$$9(x^2-8x+color{red}{16-16})-4y^2=0$$

$$9(x-4)^2-144-4y^2=0$$

so $$9(x-4)^2-4y^2=144;;;;/:144$$

$${(x-4)^2over 16}-{y^2over 36}=1$$

share|cite|improve this answer

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I believe the standard form of a hyperbola involves fractions. I believe the variables are placed as follows: ((x-h)/a^2)-((y-k)/b^2). I may have switched h and k.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

$$9(x^2-8x)-4y^2=9(x-4)^2-144-4y^2=0$$
$$iff frac{9}{144}(x-4)^2-frac{4}{144}y^2=1$$
$$iff frac{(x-4)^2}{16}-frac{y^2}{36}=1$$
$$iff frac{(x-4)^2}{4^2}-frac{y^2}{6^2}=1$$

share|cite|improve this answer

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

$endgroup$

add a comment | 

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4

$begingroup$

Note that $dfrac{(x-h)^2}{a^2}-dfrac{(y-k)^2}{b^2}=1$ is the standard form of hyperbola.

$$9x^2-4y^2-72x=0$$
$$9(x^2-8x)-4y^2=0$$
$$(x^2-8x)-dfrac49y^2=0$$
$$dfrac14(x^2-8x)-dfrac19y^2=0$$
$$dfrac14(x^2-8x+16)-dfrac19y^2=dfrac{1}{4}(16)$$
$$dfrac14(x-4)^2-dfrac19y^2=4$$
$$dfrac{(x-4)^2}{16}-dfrac{y^2}{36}=1$$
$$dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}=1mbox{ is the required Hyperbola}$$

share|cite|improve this answer

answered 2 hours ago

Key FlexKey Flex

8,63761233

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it not the equation before your answer that is in standard form since the 4^2 and 6^2 become 16 and 36. The equation with 16 & 36 as denominators.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @James $dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}$ is in the standard form.
  $endgroup$
  – Key Flex
  1 hour ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Note that $dfrac{(x-h)^2}{a^2}-dfrac{(y-k)^2}{b^2}=1$ is the standard form of hyperbola.

$$9x^2-4y^2-72x=0$$
$$9(x^2-8x)-4y^2=0$$
$$(x^2-8x)-dfrac49y^2=0$$
$$dfrac14(x^2-8x)-dfrac19y^2=0$$
$$dfrac14(x^2-8x+16)-dfrac19y^2=dfrac{1}{4}(16)$$
$$dfrac14(x-4)^2-dfrac19y^2=4$$
$$dfrac{(x-4)^2}{16}-dfrac{y^2}{36}=1$$
$$dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}=1mbox{ is the required Hyperbola}$$

share|cite|improve this answer

answered 2 hours ago

Key FlexKey Flex

8,63761233

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it not the equation before your answer that is in standard form since the 4^2 and 6^2 become 16 and 36. The equation with 16 & 36 as denominators.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @James $dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}$ is in the standard form.
  $endgroup$
  – Key Flex
  1 hour ago
  

  
  
  
  

add a comment | 

$begingroup$

Note that $dfrac{(x-h)^2}{a^2}-dfrac{(y-k)^2}{b^2}=1$ is the standard form of hyperbola.

$$9x^2-4y^2-72x=0$$
$$9(x^2-8x)-4y^2=0$$
$$(x^2-8x)-dfrac49y^2=0$$
$$dfrac14(x^2-8x)-dfrac19y^2=0$$
$$dfrac14(x^2-8x+16)-dfrac19y^2=dfrac{1}{4}(16)$$
$$dfrac14(x-4)^2-dfrac19y^2=4$$
$$dfrac{(x-4)^2}{16}-dfrac{y^2}{36}=1$$
$$dfrac{(x-4)^2}{4^2}-dfrac{(y-0)^2}{6^2}=1mbox{ is the required Hyperbola}$$

share|cite|improve this answer

answered 2 hours ago

Key FlexKey Flex

8,63761233

$endgroup$

share|cite|improve this answer

answered 2 hours ago

Key FlexKey Flex

8,63761233

answered 2 hours ago

Key FlexKey Flex

8,63761233

answered 2 hours ago

Key FlexKey Flex

8,63761233

2

$begingroup$

So we have $$9(x^2-8x)-4y^2=0$$

$$9(x^2-8x+color{red}{16-16})-4y^2=0$$

$$9(x-4)^2-144-4y^2=0$$

so $$9(x-4)^2-4y^2=144;;;;/:144$$

$${(x-4)^2over 16}-{y^2over 36}=1$$

share|cite|improve this answer

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I believe the standard form of a hyperbola involves fractions. I believe the variables are placed as follows: ((x-h)/a^2)-((y-k)/b^2). I may have switched h and k.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  

add a comment | 

2

$begingroup$

So we have $$9(x^2-8x)-4y^2=0$$

$$9(x^2-8x+color{red}{16-16})-4y^2=0$$

$$9(x-4)^2-144-4y^2=0$$

so $$9(x-4)^2-4y^2=144;;;;/:144$$

$${(x-4)^2over 16}-{y^2over 36}=1$$

share|cite|improve this answer

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I believe the standard form of a hyperbola involves fractions. I believe the variables are placed as follows: ((x-h)/a^2)-((y-k)/b^2). I may have switched h and k.
  $endgroup$
  – James
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

So we have $$9(x^2-8x)-4y^2=0$$

$$9(x^2-8x+color{red}{16-16})-4y^2=0$$

$$9(x-4)^2-144-4y^2=0$$

so $$9(x-4)^2-4y^2=144;;;;/:144$$

$${(x-4)^2over 16}-{y^2over 36}=1$$

share|cite|improve this answer

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

$endgroup$

share|cite|improve this answer

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

answered 2 hours ago

Maria MazurMaria Mazur

48k1260120

1

$begingroup$

$$9(x^2-8x)-4y^2=9(x-4)^2-144-4y^2=0$$
$$iff frac{9}{144}(x-4)^2-frac{4}{144}y^2=1$$
$$iff frac{(x-4)^2}{16}-frac{y^2}{36}=1$$
$$iff frac{(x-4)^2}{4^2}-frac{y^2}{6^2}=1$$

share|cite|improve this answer

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

$endgroup$

add a comment | 

1

$begingroup$

$$9(x^2-8x)-4y^2=9(x-4)^2-144-4y^2=0$$
$$iff frac{9}{144}(x-4)^2-frac{4}{144}y^2=1$$
$$iff frac{(x-4)^2}{16}-frac{y^2}{36}=1$$
$$iff frac{(x-4)^2}{4^2}-frac{y^2}{6^2}=1$$

share|cite|improve this answer

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

$endgroup$

add a comment | 

$begingroup$

$$9(x^2-8x)-4y^2=9(x-4)^2-144-4y^2=0$$
$$iff frac{9}{144}(x-4)^2-frac{4}{144}y^2=1$$
$$iff frac{(x-4)^2}{16}-frac{y^2}{36}=1$$
$$iff frac{(x-4)^2}{4^2}-frac{y^2}{6^2}=1$$

share|cite|improve this answer

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

$endgroup$

share|cite|improve this answer

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929

answered 2 hours ago

HAMIDINE SOUMAREHAMIDINE SOUMARE

1,20929