Tag: mathematics and statistics

Quizzes (38) Questions (955)

Questions Related to mathematics and statistics

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

For the hyperbola $\dfrac{x^2}{49}-\dfrac{y^2}{25}=1$, the equation of auxillary circle is

  1. $x^2+y^2=49$
  2. $x^2+y^2=25$
  3. $x^2+y^2=10$
  4. $x^2+y^2=10074$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $x^2+y^2=a^2$

Here the given hyperbola is $\dfrac{x^2}{49}-\dfrac{y^2}{25}=1$, which is similar to standard form of the hyperbola.

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

The equation of Auxiliary circle for the given hyperbola will also be $x^2 + y^2 = (7)^2$

$\Rightarrow x^2 + y^2 = 49$

So the correct option is $A$

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

The radius of the director circle of the ellipse $9{x^2} + 25{y^2} - 18x - 100y - 116 = 0$ is 

  1. $\sqrt {34} \,$
  2. $\sqrt {29} \,\,\,$

  3. 5

  4. 8
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is 9(x^2 - 2x) + 25(y^2 - 4y) = 116. Completing the square: 9(x-1)^2 + 25(y-2)^2 = 116 + 9 + 100 = 225. Dividing by 225 gives (x-1)^2/25 + (y-2)^2/9 = 1. The director circle radius is sqrt(a^2 + b^2) = sqrt(25 + 9) = sqrt(34).

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

The  equation of  auxillary circle of  $\dfrac{x^2}{64}-\dfrac{y^2}{36}=1$ is

  1. $x^2+y^2=100$
  2. $x^2+y^2=50$
  3. $x^2+y^2=64$
  4. $x^2+y^2=36$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $x^2+y^2=a^2$

Here the given hyperbola is $\dfrac{x^2}{64}-\dfrac{y^2}{36}=1$, which is similar to standard form of the hyperbola.

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

The equation of Auxiliary circle for the given hyperbola will also be $x^2 + y^2 = (8)^2$

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

So the correct option is $C$

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

For the hyperbola $\dfrac{x^2}{15}-\dfrac{y^2}{10}=1$, the equation of auxillary circle is

  1. $x^2+y^2=15$
  2. $x^2+y^2=10$
  3. $x^2+y^2=35$
  4. $x^2+y^2=5$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $x^2+y^2=a^2$

Here the given hyperbola is $\dfrac{x^2}{15}-\dfrac{y^2}{10}=1$, which is similar to standard form of the hyperbola.

Here $a = \sqrt{15}$ and $b = \sqrt{10}$

The equation of Auxiliary circle for the given hyperbola will also be $x^2 + y^2 = (\sqrt{15})^2$

$\Rightarrow x^2 + y^2 = 15$

So the correct option is $A$

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

 The equation of auxillary circle is $\dfrac{x^2}{25}-\dfrac{y^2}{16}=1$

  1. $x^2+y^2=16$
  2. $x^2+y^2=32$
  3. $x^2+y^2=25$
  4. $x^2+y^2=41$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $x^2+y^2=a^2$

Here the given hyperbola is $\dfrac{x^2}{25}-\dfrac{y^2}{16}=1$, which is similar to standard form of the hyperbola.

Here $a = 5$ and $b = 4$

The equation of Auxiliary circle for the given hyperbola will also be $x^2 + y^2 = (5)^2$

$\Rightarrow x^2 + y^2 = 25$

So the correct option is $C$

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

If the chords of contact of tangents drawn from $P$ to the hyperbola $x^2 - y^2 = a^2$ and its auxiliary circle are at right angle, then $P$ lies on :

  1. $x^2 - y^2 = 3a^2$
  2. $x^2 - y^2 = 2a^2$
  3. $x^2 - y^2 = 0$
  4. $x^2 - y^2 = 1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $P$ be $(h,k)$
Now Chord of contact of tangent from $P$ to the hyperbola $x^2-y^2=a^2$ is,
$T =0\Rightarrow hx -ky = a^2$ (i)
And director circle of given hyperbola is, $x^2+y^2=a^2$
Thus equation of chord of contact to this circle from P is, $hx+ky = a^2$ (ii)
Now given line (i) and (ii) are perpendicular,
$\Rightarrow \cfrac{h}{k}\times \cfrac{-h}{k}=-1\Rightarrow h^2=k^2$
Hence locus of $P$ is given by, $x^2-y^2=0$ 

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

If the circle $x^2\, +\, y^2\, =\, a^2$ intersects the hyperbola $xy\, =\, c^2$ in four points $P\, (x _1,\, y _1),\, Q(x _2,\, y _2),\, R(x _3,\, y _3),\, S(x _4,\, y _4)$, then -

  1. $X _1\, +\, X _2\, +\, X _3\, +\, X _4\, =\, 0$
  2. $Y _1\, +\, Y _2\, +\, Y _3\, +\, Y _4\, =\,0$
  3. $X _1\, X _2\, X _3\, X _4\, =\, c^4$
  4. $Y _1\,Y _2\,Y _3\,Y _4\, =\, c^4$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

Since, the circle $x^2\, +\, y^2\, =\, a^2$ intersects the hyperbola $xy\, =\, c^2$
Therefore, $x^2+\dfrac{c^4}{x^2}=a^2$
$\Rightarrow x^4-a^2x^2+c^4=0$
now sum of the roots: $x _1+x _2+x _3+x _4=0$
and product of the roots $x _1x _2x _3x _4=c^4$
Repeat the same for $y$

Multiple choice mathematics and statistics hyperbola auxiliary circle auxiliary circle, director circle director circle and auxiliary circle of a 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?

  1. $(5, 0)$
  2. $(-5, 0)$
  3. $(0, 4)$
  4. $(0, -4)$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $(-a e,0)=(-5,0)$