Tag: director circle and auxiliary circle of a hyperbola

Questions Related to director circle and auxiliary circle of a hyperbola

For the hyperbola $\dfrac{x^2}{64}-\dfrac{y^2}{36}=1$, the equation of director circle is 

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

  1. $x^2+y^2=100$

  2. $2x^2+2y^2=100$

  3. $x^2+y^2=28$

  4. $x^2-y^2=100$

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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 $\dfrac{x^2}{64} -\dfrac {y^2}{36} = 1$,

Here $a =8$ and $b =6$

So equation of the director circle will be $x^2 + y^2 = (8)^2 - (6)^2$

$\Rightarrow x^2 + y^2 = 28$

Correct option is $C$.

The equation of auxillary circle of hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$

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

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

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

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Correct Option: A
Explanation:

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


The circle drawn taken major axis as a diameter also called the 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 standard Hyperbola will be $x^2+y^2=a^2$

So the correct option is $A$

 The equation of director circle of $\dfrac{x^2}{64}-\dfrac{y^2}{49}=1$ is

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

  1. $x^2+y^2=15$

  2. $x^2+y^2=64$

  3. $x^2+y^2=18$

  4. $x^2+y^2=10$

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Correct Option: A
Explanation:

Equation of hyperbola is $\dfrac { { x }^{ 2 } }{ 64 } -\dfrac { { y }^{ 2 } }{ 49 } =1$

Here $a=8,b=7$
Equation of director circle for hyperbola is
${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }-{ b }^{ 2 }\ { x }^{ 2 }+{ y }^{ 2 }=64-49\ { x }^{ 2 }+{ y }^{ 2 }=15$
Hence, option A is correct.

The length of diameter of director circle of hyperbola $\dfrac{x^2}{49}-\dfrac{y^2}{25}=1$, is 

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

  1. $4$

  2. $6$

  3. $4\sqrt6$

  4. $24$

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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 $\dfrac{x^2}{49} -\dfrac {y^2}{25} = 1$,

Here $a =7$ and $b =5$

So equation of the director circle will be $x^2 + y^2 = (7)^2 - (5)^2$

$\Rightarrow x^2 + y^2 = 24$

Hence the radius of the director circle is $\sqrt{24} = 2\sqrt{6}$. So the diameter will be $4\sqrt6$.

So correct option is $C$.

The equation of director circle for $\dfrac{x^2}{100}-\dfrac{y^2}{36}=1$, is

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

  1. $2x^2+2y^2=100$

  2. $\sqrt 2x^2+\sqrt 2y^2=100$

  3. $x^2+y^2=6$

  4. $x^2+y^2=64$

Show answer
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 $x^2 + y^2 = a^2 - b^2$


Here the given hyperbola is $\dfrac{x^2}{100} -\dfrac {y^2}{36} = 1$,

Here $a =10$ and $b =6$

So equation of the director circle will be $x^2 + y^2 = (10)^2 - (6)^2$

$\Rightarrow x^2 + y^2 = 64$

Correct option is $D$.

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

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

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

  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: 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 $x^2 + y^2 = a^2 - b^2$


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

So correct option is $D$.

The circle passing through the vertices of hyperbola is called 

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

  1. director circle

  2. auxillary circle

  3. nine point circle

  4. none

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Correct Option: B
Explanation:

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


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 standard Hyperbola will be $x^2+y^2=a^2$

As center of the Auxiliary circle is same as the center of the  hyperbola i.e. origin and the diameter is $2a$, hence the circle touches the two vertices of hyperbola $(a,0)$ and $(-a,0)$

Hence we can say that the circle passing through the two vertices of the hyperbola is Auxiliary circle. 
So the correct option is $B$.

The intersection point of,a perpendicular on tangent of a hyperbola from the focus  and a tangent lies on 

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

  1. director circle

  2. auxillary circle

  3. nine point circle

  4. none

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Correct Option: B
Explanation:
The intersection point of a perpendicular on tangent of a hyperbola from the focus and a tangent lies on: Auxiliary circle
Auxiliary circle of a hyperbola which is a circle described on the major axis of a hyperbola as its diameter.
Let the hyperbola be
$\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$
The equation of auxiliary circle is $x^2+y^2=a^2$
Take a point $P(x _1,y _1)$
Through $P$ draw a line perpendicular to major axis intersecting major axis in $N$ and auxiliary circle in $P'$.
The points $P$ and $P'$ are called as corresponding points on the hyperbola and auxiliary circle respectively.
This angle is known as the eccentric angle of the point $P$ on the hyperbola and auxiliary circle respectively.

If $\theta$ is eliminated from the equations $a\sec\theta - x\tan\theta = y \mbox{ and } b\sec\theta + y\tan\theta = x$ ($a$ and $b$ are constant), then the eliminant denotes the equation of 

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

  1. the director circle of the hyperbola $\displaystyle\frac{x^2}{a^2} - \displaystyle\frac{y^2}{b^2} = 1$

  2. auxiliary circle of the ellipse $\displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} = 1$

  3. director circle of the ellipse $\displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} = 1$

  4. director circle of the circle $x^2 + y^2 = \displaystyle\frac{a^2 + b^2}{2}$

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Correct Option: C
Explanation:

Solving given equation we get,
$\displaystyle \tan\theta=\frac{ax-by}{ay+bx}$
and $\displaystyle \sec\theta = \frac{x^2+y^2}{ay+bx}$
Eliminating $\theta$ we get, $x^2+y^2=a^2+b^2$ which is director  circle of the ellipse $\displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} = 1$

If pair of tangents are drawn from any point $(p)$ on the circle ${x^2} + {y^2} = 1$ to the hyperbola $\frac{{{x^2}}}{2} - \frac{{{y^2}}}{1} = 1$ such that locus of circumcenter of triangle formed by pair of tangents and chord of contact is ${\lambda _1}{x^2} - 2{\lambda _2}{y^2} = 2{\left( {\frac{{{x^2}}}{2} - {y^2}} \right)^2}$, then 

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

  1. ${\lambda _1} = 2,{\lambda _2} = 1$

  2. ${\lambda ^2} _1 + {\lambda ^2} _2 = 5$

  3. ${\lambda _1} = 1,{\lambda _2} = - 1$

  4. ${\lambda ^2} _1 + {\lambda ^2} _2 = 2$

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Correct Option: A