The type of conic sections for the nondegenerate equations given below.a) 6x^{2} + 3x + 10y=10y^{2} + 8b) 3x^{2} + 18xy=5x + 2y + 9c) 4x^{2} + 8x - 5= -y^{2} + 6y + 3

floymdiT 2020-10-18 Answered

The type of conic sections for the nondegenerate equations given below.

a) \(\displaystyle{6}{x}^{{{2}}}\ +\ {3}{x}\ +\ {10}{y}={10}{y}^{{{2}}}\ +\ {8}\)

b) \(\displaystyle{3}{x}^{{{2}}}\ +\ {18}{x}{y}={5}{x}\ +\ {2}{y}\ +\ {9}\)

c) \(\displaystyle{4}{x}^{{{2}}}\ +\ {8}{x}\ -\ {5}=\ -{y}^{{{2}}}\ +\ {6}{y}\ +\ {3}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Bella
Answered 2020-10-19 Author has 6365 answers

a) Consider the equation \(\displaystyle{6}{x}^{{{2}}}\ +\ {3}{x}\ +\ {10}{y}={10}{y}^{{{2}}}\ +\ {8}.\)

This equation can be written as: \(\displaystyle{6}{x}^{{{2}}}\ -\ {10}{y}^{{{2}}}\ +\ {10}{y}\ -\ {8}={0}\).

It is known that the equation is in the form of \(\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\)

Here, the coefficient of \(\displaystyle{x}^{{{2}}}\) and \(\displaystyle{y}^{{{2}}}\) are \(\displaystyle{A}={6}\) and \(\displaystyle{C}=\ -{10}\), which means \(\displaystyle{A}{C}=\ -{60},\) that is, \(\displaystyle{A}{C}\ {<}\ {0}\).

Therefore, the equation is of a hyperbola. Hence, the conic section of the given equation is a hyperbola.

b) Consider the equation \(\displaystyle{3}{x}^{{{2}}}\ +\ {18}{x}{y}={5}{x}\ +\ {2}{y}\ +\ {9}\).

This equation caan be written as: \(\displaystyle{3}{x}^{{{2}}}\ +\ {18}{x}{y}\ -\ {5}{x}\ -\ {2}{y}\ -\ {9}={0}\), which is in the form of \(\displaystyle{A}{X}^{{{2}}}\ +\ {B}{x}{y}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\).

Solve for \(\displaystyle{B}^{{{2}}}\ -\ {4}{A}{C}\)
\(\displaystyle{B}^{{{2}}}\ -\ {4}{A}{C}={\left({18}\right)}\ -\ {4}{\left({3}\right)}{\left({0}\right)}\)
\(\displaystyle={324}\ -\ {0}\)
\(\displaystyle={324}\)

This shows that \(\displaystyle{B}^{{{2}}}\ -\ {4}{A}{C}\ {>}\ {0}.\) Therefore, the equation is of a hyperbola. Hence, the conic section of the given equation is a heperbola.

c) Consider the equation \(\displaystyle{4}{x}^{{{2}}}\ +\ {8}{x}\ -\ {5}=\ -{y}^{{{2}}}\ +\ {6}{y}\ +\ {3}\). T

his equation can written as: \(\displaystyle{4}{x}^{{{2}}}\ +\ {y}^{{{2}}}\ +\ {8}{x}\ -\ {6}{y}\ -\ {8}={0}\). It is known that the equation is in the form \(\displaystyle{A}{X}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}\) Here, \(\displaystyle{A}={4}\ \text{and}\ {C}={1},\) which means \(\displaystyle{A}{C}={4}\ \text{and}\ {A}{C}\ {>}\ {0}.\)

Therefore, the equation is of an ellipse. Hence, the conic section of the given equation is an ellipse.

Have a similar question?
Ask An Expert
32
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-08-08
The type of conic sections for the nondegenerate equations given below.
a) \(\displaystyle{0.1}{x}^{{{2}}}+{0.6}{x}-{1.6}={0.2}{y}-{0.1}{y}^{{{2}}}\)
b) \(\displaystyle{2}{x}^{{{2}}}-{7}{x}{y}=-{y}^{{{2}}}+{4}{x}-{2}{y}-{1}\)
c) \(\displaystyle{8}{x}+{2}{y}={y}^{{{2}}}+{4}\)
asked 2020-11-08

The type of conic sections for the nondegenerate equations given below.

a) \(\displaystyle{8}{x}^{{{2}}}\ -\ {2}{y}^{{{2}}}\ -\ {3}{x}\ +\ {2}{y}\ -\ {6}={0} \ \)

b) \(\displaystyle-{6}{y}^{{{2}}}\ +\ {4}{x}\ -\ {12}{y}\ -\ {24}={0}\)

c) \(\displaystyle-{9}{x}^{{{2}}}\ -\ {4}{y}^{{{2}}}\ -\ {18}{x}\ +\ {12}{y}={0}\)

asked 2021-08-07
Sketch the following conic sections:
a)\(\displaystyle{y}^{{2}}-{4}{y}+{8}-{4}{x}^{{2}}={8}\)
b)\(\displaystyle{y}^{{2}}-{4}{y}-{4}{x}^{{2}}={8}\)
c)\(\displaystyle{3}{\left({y}-{5}\right)}^{{2}}-{7}{\left({x}+{1}\right)}^{{2}}={1}\)
asked 2021-08-09
Show that the equation represents a conic section. Sketch the conic section, and indicate all pertinent information (such as foci, directrix, asymptotes, and so on). \(\displaystyle{\left({a}\right)}{x}^{{2}}–{2}{x}–{4}{y}^{{2}}–{12}{y}=-{8}{\left({b}\right)}{2}{x}^{{2}}+{4}{x}-{5}{y}+{7}={0}{\left({c}\right)}{8}{a}^{{2}}+{8}{x}+{2}{y}^{{2}}–{20}{y}={12}\)
asked 2020-12-31

The conclusion about the values of e for rlliptical equations. The provided equations of conic sections are,

A) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{36}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{13}}}}={1}\)

B) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{4}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{4}}}}={1}\)

C) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{25}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{16}}}}={1}\)

D) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{25}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{39}}}}={1}\)

E) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{17}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{81}}}}={1}\)

F) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{36}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{36}}}}={1}\)

G) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{16}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{65}}}}={1}\)

H) \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{144}}}}\ -\ {\frac{{{y}^{{{2}}}}}{{{140}}}}={1}\)

asked 2021-08-15
Sketch the following conic sections:
\(\displaystyle{4}^{{{2}}}-{4}{y}+{8}-{4}{x}^{{{2}}}={0}\)
asked 2021-08-11
Provide the standard equations for lines and conic sections in polar coordinates with examples.
...