Tag: forms of equations of a hyperbola

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Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

For the hyperbola $16x^2\, -\, 9y^2\, +\, 32x\, +\, 36y\,-\, 164\, =\, 0$, find $2(a+b)$.

  1. $8$
  2. $6$
  3. $14$
  4. $12$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given hyperbola can be written as


$16x^2+32x+16-9y^2+36y-36=144$

$\displaystyle\, \frac{(x\, +\, 1)^2}{9}\, -\, \frac{(y\, -\, 2)^2}{16}\, =\, 1$

$\Rightarrow a =3, b = 4$

Length of major axis $=2\times 4=8$ and length of minor axis $=2\times 3 = 6$. 

Hence sum is $2(a+b)=2(3+4)=14.$

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

A hyperbola having the transverse axis of length $\sqrt{2}$ is confocal with $3x^2 + 4y^2 = 12$, then its equation is:

  1. $2x^2-2y^2=1$
  2. $2x^2+2y^2=1$
  3. $x^2+y^2=2$
  4. $x^2-y^2=2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given ellipse may be written as, $\displaystyle \frac {x^2}{4}+\frac {y^2}{3}=1$
$\displaystyle \Rightarrow a^2 = 4, b^2 = 3\therefore e = \sqrt{1-\frac{3}{4}}=\frac{1}{2}$
Thus foci of ellipse $(\pm 1, 0)$
Hence foci of the hyperbola is $(\pm 1, 0)$
And semi-major axis $a =\cfrac{1}{\sqrt{2}}$
$\Rightarrow \pm 1=\sqrt {a^2+b^2}\Rightarrow b^2=\frac {1}{2}$
or $b^2=1-a^2=1-\frac {1}{2}=\frac {1}{2}$
The required equation of hyperbola is,
$\displaystyle \frac {x^2}{1/2}-\frac {y^2}{1/2}=1\Rightarrow 2x^2-2y^2=1$

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

Find the equation to the hyperbola, the distance between whose foci is $16$ and whose eccentricity is $\sqrt{2}$.

  1. $x^2\, -\, y^2\, =\, 32$
  2. $x^2\, -\, y^2\, =\, 18$
  3. $x^2\, -\, y^2\, =\, 64$
  4. $x^2\, -\, y^2\, =\, 48$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given eccentricity of the hyperbola is $\sqrt{2}$
Hence hyperbola is rectangular $a=b$
Also given distance between focii is 16. $\Rightarrow 2ae = 16\Rightarrow a =4\sqrt{2}$
Hence required hyperbola is given by, $x^2-y^2=a^2=32$

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

A parabola is drawn with its vertex at $(0,-3)$, the axis of symmetry along the conjugate axis of the hyperbola $\displaystyle \frac { { x }^{ 2 } }{ 49 } -\frac { { y }^{ 2 } }{ 9 } =1$ and passing through the two foci of the hyperbola. The coordinates of the focus of the parabola are :

  1. $\displaystyle \left( 0,\frac { 11 }{ 6 } \right) $
  2. $\displaystyle \left( 0,-\frac { 11 }{ 6 } \right) $
  3. $\displaystyle \left( 0,\frac { 11 }{ 12 } \right) $
  4. $\displaystyle \left( 0,-\frac { 11 }{ 12 } \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation of hyperbola is $\displaystyle \frac { { x }^{ 2 } }{ 49 } -\frac { { y }^{ 2 } }{ 9 } =1$

Its conjugate axis is y-axis
Also, $\displaystyle e=\sqrt { 1+\frac { { b }^{ 2 } }{ { a }^{ 2 } }  } =\sqrt { 1+\frac { 9 }{ 49 }  } =\frac { \sqrt { 58 }  }{ 7 } $
$\therefore$ Foci of hyperbola is $\left( \pm ae,0 \right) \Rightarrow \left( \pm \sqrt { 58 } ,0 \right) $
Now equation of parabola with vertex at $(0,-3)$ and axis along y-axis is ${ x }^{ 2 }=l\left( y+3 \right) $
It passes through $\left( \pm \sqrt { 58 } ,0 \right) $
$\displaystyle \therefore 58=l\left( 0+3 \right) \Rightarrow l=\frac { 58 }{ 3 } $
$\therefore$ parabola is $\displaystyle { x }^{ 2 }=\frac { 58 }{ 3 } \left( y+3 \right) $
Its focus is $\displaystyle \left( 0,-3+\frac { 58 }{ 4.3 }  \right) \equiv \left( 0,\frac { 11 }{ 6 }  \right) $

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

Which of the following is true for the hyperbola $9x^2\, -\, 16y^2\, -\, 18x\, +\, 32y\, -\, 151\, =\, 0$?

  1. The length of the transverse axes is $4$
  2. Length of latus rectum is $9$
  3. Equation of directrix is $x\, =\, \displaystyle \frac{21}{5}$ and $x\, =\, - \displaystyle \frac{11}{5}$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$9x^2\, -\, 16y^2\, -\, 18x\, +\, 32y\, -\, 151\, =\, 0$
$9(x^2-2x)-16(y^2-2y)=151$
$9(x^2-2x+1)-16(y^2-2y+1)=151-7=144$
$9(x-1)^2-16(y-1)^2=144$
$\cfrac{(x-1)^2}{16}-\cfrac{(y-1)^2}{9}=1$
$\Rightarrow a^2=16, b^2=9$
$\therefore e=\sqrt{1+\dfrac{b^2}{a^2}}=\cfrac{5}{4}$

$\therefore$ The length of the transverse axis is $=2a=8$
Length of latus rectum is $=2\cfrac{b^2}{a}=\cfrac{9}{2}$
Equation of directrix is, $x=1\pm \cfrac{a}{e}=1 \pm \cfrac{16}{5}=\cfrac{21}{5}$ or $-\cfrac{11}{5}$
Hence, option 'C' is correct.

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

An ellipse intersects the hyperbola $\displaystyle 2x^{2}-2y^{2}=1$ orthogonally at point $P$. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the co-ordinate axes and product of focal distances of $P$ is $x$ then $2x$ is:

  1. $1$
  2. $2$
  3. $3$
  4. $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

As ellipse and hyperbola intersect orthogonally 
$\displaystyle \Rightarrow $ their foci are coincident
Now we have $\displaystyle SP+S,P=2a\Rightarrow $ length of major axis ellipse ..........(1)
And $\displaystyle \left | SP-S,P \right |=2a'\Rightarrow $ length of transverse axis of hyperbola ....... (2)
$\displaystyle (1)^{2}-(2)^{2}\Rightarrow 4PS'PS=4(a^{2}-a'^{2})=4\left [ \left ( \frac{a'e}{e} \right )^{2} -a'^{2}\right ]=4\left ( 2-\frac{1}{2} \right )=6$
$\displaystyle \Rightarrow 2x = 3$

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

The equations of the transverse and conjugate axes of a hyperbola are respectively $x + 2y - 3 = 0, 2x - y + 4 = 0$ and their respective lengths are $\displaystyle \sqrt{2}$ 2/$\displaystyle \sqrt{2}$. The equation of the hyperbola is 

  1. $\displaystyle \frac{2}{5}(x+2y-3)^{2}-\frac{3}{5}(2x-y+4)^{2}=1$
  2. $\displaystyle \frac{2}{5}(2x+y-4)^{2}-\frac{3}{5}(x+2y3-4)^{2}=1$
  3. $\displaystyle 2(2x-y+4)^{2}-3(x+2y-3)^{2}=1$
  4. $\displaystyle 2(2x+2y-3)^{2}-3(2x-y+4)^{2}=1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

It is given that $2a=\sqrt {2}$ which implies that $a=\dfrac { 1 }{ \sqrt { 2 }  }$.


Also, $2b=\dfrac { 2 }{ \sqrt { 3 }  } \Rightarrow b=\dfrac { 1 }{ \sqrt { 3 }  }$ 

If we take the two axes as the new coordinate system and the point of intersection of the axes of the new origin, then in the new coordinate system, equation of the hyperbola will be:

$\dfrac { { X }^{ 2 } }{ a^{ 2 } } -\dfrac { { Y }^{ 2 } }{ b^{ 2 } } =1\ \Rightarrow \dfrac { { X }^{ 2 } }{ \left( \dfrac { 1 }{ \sqrt { 2 }  }  \right) ^{ 2 } } -\dfrac { { Y }^{ 2 } }{ \left( \dfrac { 1 }{ \sqrt { 3 }  }  \right) ^{ 2 } } =1\ \Rightarrow \dfrac { { X }^{ 2 } }{ \dfrac { 1 }{ 2 }  } -\dfrac { { Y }^{ 2 } }{ \dfrac { 1 }{ 3 }  } =1\ \Rightarrow 2{ X }^{ 2 }-3{ Y }^{ 2 }=1\quad ....(1)$

Let $P(x,y)$ be the coordinates of a point on the hyperbola in original x-y system, then 

$X=\dfrac { \left| 2x-y+4 \right|  }{ \sqrt { 5 }  } ,\quad Y=\dfrac { \left| x+2y-3 \right|  }{ \sqrt { 5 }  } $

($\because$ $X$ is the distance of a point on hyperbola from $2x-y+4=0$ and $Y$ is the distance of a point on hyperbola from $x+2y-3=0$)

Therefore, equation 1 becomes:

$2{ \left( \dfrac { \left| 2x-y+4 \right|  }{ \sqrt { 5 }  }  \right)  }^{ 2 }-3{ \left( \dfrac { \left| x+2y-3 \right|  }{ \sqrt { 5 }  }  \right)  }^{ 2 }=1\ \Rightarrow \dfrac { 2 }{ 5 } { \left( 2x-y+4 \right)  }^{ 2 }-\dfrac { 3 }{ 5 } { \left( x+2y-3 \right)  }^{ 2 }=1$

Hence, the equation of the hyperbola is $\dfrac { 2 }{ 5 } { \left( 2x-y+4 \right)  }^{ 2 }-\dfrac { 3 }{ 5 } { \left( x+2y-3 \right)  }^{ 2 }=1$.

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

MATCH THE FOLLOWING
Hyperbola                                                   Length of latusrectum
A}$x^{2}-4y^{2}=4$                                               1. 1
B}$25x^{2}-16y^{2}=400$                                     2.12
C}$ 2x^{2}-y^{2}-4x-4y-20=0$                   3.9/2
D)$9x^{2}-16y^{2}+72x-32y-16=0$           4. 25/2

The correct match is

  1. I II III IV

    1 2 3 4

  2. 1 4 2 3

  3. 3 1 2 4

  4. 2 3 4 1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(A) $\dfrac{x^{2}}{4}-y^{2}=1$


$LR=\dfrac{2b^{2}}{a}$

$=\dfrac{2}{2}$

$=1$

(B)$\dfrac{x^{2}}{16}-\frac{y^{2}}{25}=1$

$b=5, a=4$

$LR=\dfrac{2.25}{4}$

$=\dfrac{25}{2}$

(C) $2(x-1)^{2}-(y-2)^{2}=18$

$\dfrac{(x-1)^{2}}{9}-\frac{(y-2)^{2}}{18}=1$

$b^{2}=18, a=3$

$LR=\dfrac{2\times 18}{3}$

$=12$

(D) $\dfrac{(x+4)^{2}}{16}-\frac{(y+1)^{2}}{9}=1$

$b^{2}=9$

$a=4$

$LR=2\times \dfrac{9}{4}$

$=\dfrac {9}{2}$

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

The equation to the hyperbola having its eccentricity $2$ and the distance between its foci is $8$, is

  1. $\dfrac {x^{2}}{12} - \dfrac {y^{2}}{4} = 1$
  2. $\dfrac {x^{2}}{4} - \dfrac {y^{2}}{12} = 1$
  3. $\dfrac {x^{2}}{8} - \dfrac {y^{2}}{2} = 1$
  4. $\dfrac {x^{2}}{16} - \dfrac {y^{2}}{9} = 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the equation of hyperbola is $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1$
Given, $e = 2, 2ae = 8$
$\Rightarrow ae = 4\Rightarrow a = 2$
Now, $b^{2} = a^{2} (e^{2} - 1)$
$\Rightarrow b^{2} = 4(4- 1)$
$\Rightarrow b^{2} = 12$
$\therefore$ Equation of hyperbola is
$\dfrac {x^{2}}{4} - \dfrac {y^{2}}{12} = 1$