Tag: hyperbola

Questions Related to hyperbola

If one of the directrix of hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{b}=1$ is $x=-\dfrac{9}{5}$. Then the corresponding focus of hyperbola is?

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $(5, 0)$

  2. $(-5, 0)$

  3. $(0, 4)$

  4. $(0, -4)$

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Correct Option: B
Explanation:

Given hyperbola is $\dfrac{x^2}{9}-\dfrac{y^2}{b}=1$

Directrix is $x=-\dfrac{9}{5}\implies \dfrac{3}{e}=\dfrac{9}{5}\implies e=\dfrac{5}{3}$
Corresponding focus of hyperbola is $(-a e,0)=(-5,0)$

The equation of the director circle of the hyperbola $\dfrac{x^2}{81}- \dfrac{y^2}{16}=1$ is

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=65$

  2. $x^2+y^2=97$

  3. $(x-9)^2+(y-4)^2=0$

  4. $(x+9)^2+(y+4)^2=0$

Show answer
Correct Option: A
Explanation:

for hyperbola,

$\dfrac{x^2}{81}- \dfrac{y^2}{16}=1$ 
equation of director circle is 
$x^2+y^2=a^2-b^2$
$x^2+y^2=81-16=65$

The equation of the director circle of the hyperbola $\dfrac{x^2}{36}- \dfrac{y^2}{16}=1$ is

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=20$

  2. $x^2+y^2=52$

  3. $(x-9)^2+(y-4)^2=0$

  4. $(x+9)^2+(y+4)^2=0$

Show answer
Correct Option: A
Explanation:

for hyperbola, $\dfrac{x^2}{36}- \dfrac{y^2}{16}=1$ 

equation of director circle is $x^2+y^2=a^2-b^2=36-16=20$

Auxiliary circle of a hyperbola is defined as:

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. The auxiliary circle for a hyperbola is a circle with its centre on the polar and contains the two vertices.

  2. The circle whose center concurs with that of the ellipse and whose radius is equal to the ellipse's semimajor axis.

  3. The auxiliary circle for a hyperbola is a circle with its centre on the axis and contains the two vertices.

  4. None of these

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Correct Option: C
Explanation:

The Auxiliary circle of a hyperbola is defined as the circle with the center same as the hyperbola and with transverse axis as it's diameter. The end points of transverse axis are the two vertices of the hyperbola, so the circle also contains the two vertices of the hyperbola.


The equation of Auxiliary circle foe a hyperbola is given by $x^2  + y^2 = a^2$

Hence the correct answer is Option $C$.

The circle with major axis as diameter is called the auxiliary circle of the hyperbola. 
If $a>b,$ then the equation of auxiliary circle is

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=a^2$

  2. $x^2+y^2=b^2$

  3. $x^2+y^2=a^2-b^2$

  4. $x^2+y^2=a^2+b^2$

Show answer
Correct Option: A
Explanation:

For any Hyperbola of the form $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$


Length of Major axis is always $2a$, though $a$ can be greater, lesser or equal to $b$

For any conjugate Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=-1$ or $\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1$

The length of major axis is always $2b$, Though $b$  can be greater, lesser or equal to $a$

So For any Hyperbola the circle drawn taken major axis as a diameter also called auxiliary circle of the Hyperbola will have a diameter of $2a$, equal to the length of major axis and center same as center of Hyperbola.

Hence equation of Auxiliary circle of any Hyperbola will be $x^2+y^2=a^2$

So the correct option is $A$

The equation of director circle of the hyperbola $-\dfrac{x^2}{a^2}+ \dfrac{y^2}{b^2}=1$, if $b>a$,  is 

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=b^2-a^2$

  2. $x^2+y^2=b^2$

  3. $x^2+y^2=a^2$

  4. $x^2+y^2=b^2+a^2$

Show answer
Correct Option: A
Explanation:

The equation of director circle whose center us at origin and radius $\sqrt{a^2-b^2}$

So, $x^2+y^2=a^2-b^2$
If $b>a$, then the equation of director circle of the hyperbola will be  $x^2+y^2=b^2-a^2$, where radius will be $\sqrt{b^2-a^2}$.

The radius of director circle of the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ is

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $6$

  2. $7$

  3. $\sqrt 7$

  4. $8$

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Correct Option: C
Explanation:

The Director circle of a hyperbola is defined as the locus of the point of intersection of two perpendicular tangents to the hyperbola. For any standard hyperbola $\dfrac{x^2}{a^2} -\dfrac {y^2}{b^2} = 1$,


The equation of Director circle is given by $x^2 + y^2 = a^2 - b^2$


Here the given hyperbola is $\dfrac{x^2}{16} -\dfrac {y^2}{9} = 1$,

Here $a =4$ and $b =3$

So equation of the director circle will be $x^2 + y^2 = (4)^2 - (3)^2$

$\rightarrow x^2 + y^2 = 7$

Hence the radius of the director circle is $\sqrt7$. So correct option is $C$.

The equation of director circle of $-\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, If $b<a$ is:

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=b^2-a^2$

  2. $x^2+y^2=b^2+a^2$

  3. $x^2-y^2=b^2-a^2$

  4. Director circle does not exist

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Correct Option: D
Explanation:

The Director circle of a hyperbola is defined as the locus of the point of intersection of two perpendicular tangents to the hyperbola. For any standard hyperbola $\dfrac{x^2}{a^2} -\dfrac {y^2}{b^2} = 1$,


The equation of Director circle is given by $x^2 + y^2 = a^2 - b^2$

The given hyperbola $-\dfrac{x^2}{a^2} +\dfrac {y^2}{b^2} = 1$ is a conjugate hyperbola. 

Hence the equation of Director circle for a conjugate hyperbola is given by $x^2 + y^2 = b^2 - a^2$


Hence the Director circle of given hyperbola is a circle whose center is same as center of the given hyperbola and the radius is $\sqrt{b^2 - a^2}$

As the radius is always a positive and real value, so $(b^2 - a^2) > 0$

$\rightarrow (b-a)(b+a) > 0$

As $a$ and $b$ both are positive quantities hence $a + b >0$

Hence $b-a > 0$  or  $b > a$

For $b < a$ the director circle does not exist, as the radius will not be real for $b<a$

So correct option is $D$.

The equation of the auxiliary circle of the hyperbola $4x^2-9y^2=36$ is

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $x^2+y^2=81$

  2. $x^2+y^2=9$

  3. $x^2+y^2=16$

  4. $x^2+y^2=4$

Show answer
Correct Option: B
Explanation:

for hyperbola $a > b$

equation of auxiliary circle is $x^2+y^2=a^2$
$x^2+y^2=9$ is the equation of auxiliary circle for the given hyperbola

If any tangent to the hyperbola  $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ with centre $C$, meets its director circle in $P$ and $Q$, then:

mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a hyperbola

  1. $CP$ and $CQ$ are perpendicular to each other.

  2. $CP$ and $CQ$ are conjugate semi-diameters of the hyperbola.

  3. $CP$ and $CQ$ are not conjugate semi-diameters of the hyperbola.

  4. None of the above

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Correct Option: B