Question

How do u find the center, vertices, foci and asymptotes of $ \dfrac{{{x^2}}}{7} - \dfrac{{{y^2}}}{9} = 1 $ ?

Answer 1

Hint: We should know about the equation of hyperbola before stat solving the question.
The standard form of equation of a hyperbola with center $ (c,d) $ and transverse axis on the x-axis.
 \[\dfrac{{{{(x - c)}^2}}}{{{a^2}}} - \dfrac{{{{(y - d)}^2}}}{{{b^2}}} = 1\]
Where,
The length of the transverse axis is $ 2a $ .
And, the length of the conjugate axis is the conjugate axis.

Complete step-by-step answer:
Step 1:
We make given equation simpler for us by writing in square of number to denominator values:
  $ \dfrac{{{x^2}}}{{\sqrt 7 }} - \dfrac{{{y^2}}}{{{3^2}}} = 1 $
Step 2:
Compare it with standard equation:
 \[\dfrac{{{{(x - c)}^2}}}{{{a^2}}} - \dfrac{{{{(y - d)}^2}}}{{{b^2}}} = 1\]
We get,
Centre $ C = (0,0) $
The vertices are $ V' = ( - a,0) = ( - \sqrt 7 ,0) $ and \[V' = (a,0) = (\sqrt 7 ,0)\] .
To calculate the foci, we need the distance from the centre to the foci,
  $ {c^2} = {a^2} + {b^2} $
  $ \Rightarrow {c^2} = 7 + 9 = 16 $
  $ \Rightarrow c = \pm 4 $
The foci are $ F' = ( - c,0) = ( - 4,0) $ and $ F = (c,0) = (4,0) $ .
The asymptotes will be,
  $ \dfrac{{{x^2}}}{{\sqrt 7 }} - \dfrac{{{y^2}}}{{{3^2}}} = 1 $
  $ y = \pm \dfrac{3}{{\sqrt 7 }} $ .

Note: As we come through many applications of hyperbola in our daily life. A guitar is an example of a hyperbola as its sides form a hyperbola. Airport design is hyperbolic parabolic. As it has one cross section of hyperbola and others have a parabola. Gear transmission having pair of hyperbolic gear. Hyperbola is important in astronomy as they are the path followed by the non-recurrent comets.