Tag: mathematics and statistics

Questions Related to mathematics and statistics

The asymptote of the hyperbole $\dfrac {x^{2}}{a^{2}-y^{2}b^{2}}=1$ from with any tangent to the hyperbola a triangle whose area is $a^{2}tan\lambda$ in magnitude then its eccentricity is ?

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $Sec\lambda$

  2. $csc\lambda$

  3. $sec^{2}\lambda$

  4. $csc^{2}\lambda$

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Correct Option: C

Differential equation of all hyperbolas which pass through the origin, and have their asymptotes parallel to the coordinate axes is?

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $xy\dfrac{d^2y}{dx^2}-2x\left(\dfrac{dy}{dx}\right)^2+2y=0$

  2. $xy\dfrac{d^2y}{dx^2}-2\left(\dfrac{dy}{dx}\right)^2+2y\left(\dfrac{dy}{dx}\right)=0$

  3. $xy\left(\dfrac{d^2y}{dx^2}\right)-2x\left(\dfrac{dy}{dx}\right)^2+2y\dfrac{dy}{dx}=0$

  4. $xy\dfrac{d^2y}{dx^2}+2x\left(\dfrac{dy}{dx}\right)^2+y\left(\dfrac{dy}{dx}\right)=0$

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Correct Option: A

Area of triangle formed by the tangent at one vertex and asymptotes of the hyperbola xy=2

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. 2sq. units

  2. 3 units

  3. 1 sq. unit

  4. none of these

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Correct Option: A

The product of perpendiculars drawn from any point of a hyperbola with principal axes $2a$ and $2b$ upon its asymptotes is equal to:

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $\frac{a^2b^2}{a^2+b^2}$

  2. $\frac{a^2 +b^2}{a^2b^2}$

  3. $\frac{ab}{a^2+b^2}$

  4. $\frac{ab(a+b)}{\sqrt a+\sqrt b}$

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Correct Option: A

The angle between the asymptotes of the hyperbola $24x^2 - 8y^2 = 27$ is 

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $90^o$

  2. $60^o$

  3. $120^o$

  4. $45^o$

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Correct Option: C
Explanation:
$24x^{2}-8y^{2}=27$

divide the above equation with 27 
$\displaystyle \frac{n^{2}}{\dfrac{27}{24}}=\frac{y^{2}}{\dfrac{27}{8}}=1$

$\displaystyle \frac{n^{2}}{\dfrac{9}{8}}-\frac{y^{2}}{\dfrac{27}{8}}=1$

$\displaystyle a^{3}=\frac{9}{8}, b^{2}=\frac{27}{8}$

$\displaystyle a=\frac{3}{2\sqrt{2}},b=\frac{3\sqrt{3}}{2\sqrt{2}}$

let $ 2\alpha $ be the angle between asymptotes 

$2\alpha =2\tan^{-1}\dfrac{b}{a}$

$\displaystyle =2\tan^{-1}\frac{\frac{3\sqrt{3}}{2\sqrt{2}}}{\frac{3}{2\sqrt{2}}}$

$=2\tan^{-1}\sqrt{3}$

$= 2\times \dfrac{\pi}{3}$

$\displaystyle =\frac{2\pi }{3}$  or  $120$
Evaluate the following definite integral:
$\displaystyle \int _{0}^1 \dfrac {1-x}{1+x}dx$
mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $2\log 2+1$

  2. $2\log 2$

  3. $2\log 2-1$

  4. $\log 4$

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Correct Option: C
Explanation:

$I=\displaystyle \int _{0}^1 \dfrac {1-x}{1+x}dx$ 

$=\displaystyle \int _{0}^1 \dfrac {2-(1+x)}{1+x} dx$ 

$=\displaystyle \int _{0}^1 \left (\dfrac {2}{1+x} -1 \right)dx $

$=\left [2\log (x+1)-x \right] _0^1$

$\Rightarrow \ I=(2\log 2-1)- (2\log 1-0)$ 

$=2\log 2-1$

If a line intersect a hyperbola at $(-2,-6)$ and $(4,2)$ and one of the asymtote at $(1,-2)$, then the centre of the hyperbola is

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $(7,6)$

  2. $(1,-2)$

  3. $(10,10)$

  4. $(-5,-10)$

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Correct Option: A

Let product of distances of any point hyperbola (x+y-1) (x-y+3)= 60 to its asymptotes is 'K' then K is divisible by

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. 2

  2. 3

  3. 4

  4. 5

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Correct Option: A

If the cordinate of any point p on the hyperbola $9{x^2} - 16{y^2} = 144$ is produced to cut the asymptotes in the points Q and R. Then the product PQ.PR equals to:

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. $9$

  2. $\dfrac{12}{5} $

  3. $\dfrac{144}{25}$

  4. $7$

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Correct Option: C
Explanation:
$\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$

Asymplote is $y=\pm\dfrac{3}{4}x$

Let us take $4y=3x, 4y=-3x$

consider a parametric point $(4\sec\theta, 3\tan\theta)$ on the parabola

$Q$ is intersection with $4y=3x$

then $PQ=\left|\dfrac{12\sec\theta-12\tan\theta}{\sqrt{4^{2}+3^{2}}}\right|$

$PQ=\left|\dfrac{12}{5}(\sec \theta-\tan\theta)\right|$

$R$ is intersection with $4y=-3x$

then $PR=\left|\dfrac{12\sec\theta+12\tan\theta}{\sqrt{4^{2}+3^{2}}}\right|$

$PQ.PR=\dfrac{144}{25}(\sec\theta+\tan\theta)(\sec\theta-\tan\theta)$

$=\dfrac{144}{25}(\sec^{2}\theta+\tan^{2}\theta)=\dfrac{144}{25}(1)$

`e`$=\dfrac{144}{25}$

The points of intersection of asymptotes with directrices lies on

mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes

  1. Auxillary circle

  2. Director circle

  3. Transverse axis

  4. Conjugate axis

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Correct Option: A
Explanation:
  • Asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through the extremities of each axis parallel to the other axis.
  • A perpendicular drawn from the foci on either asymptote meet it in the same points as the corresponding directrix and the common points of intersection lie on the auxiliary circle.