Tag: asymptotes of a curve

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Questions Related to asymptotes of a curve

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

The angle between the asymptotes to the hyperbola $\dfrac { { x }^{ 2 } }{ 16 } -\dfrac { { y }^{ 2 } }{ 9 } =1$ is

  1. $\pi -2\tan ^{ -1 }{ \left( \dfrac { 3 }{ 4 } \right) } $
  2. $\pi -2\tan ^{ -1 }{ \left( \dfrac { 4 }{ 3 } \right) } $
  3. $2\tan ^{ -1 }{ \left( \dfrac { 3 }{ 4 } \right) } $
  4. $2\tan ^{ -1 }{ \left( \dfrac { 4 }{ 3 } \right) } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Fact: Angle between asymptotes of hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is $2\tan^{-1}\left(\dfrac{b}{a}\right)$


Hence angle between the asymptotes to the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ is $2\tan^{-1}\left(\dfrac{3}{4}\right)$

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

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

  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. $xy\dfrac{d^2y}{dx^2}+2x\left(\dfrac{dy}{dx}\right)^2+y\left(\dfrac{dy}{dx}\right)=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Hyperbolas with asymptotes parallel to axes have the form (x-h)(y-k) = c. Differentiating twice leads to the differential equation xy(d^2y/dx^2) - 2x(dy/dx)^2 + 2y(dy/dx) = 0 (or similar depending on form). Option A is the standard differential equation for this family.

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

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

  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. $\frac{ab(a+b)}{\sqrt a+\sqrt b}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of the perpendiculars from any point on the hyperbola to its asymptotes is a^2b^2 / (a^2 + b^2).

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

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

  1. $90^o$
  2. $60^o$
  3. $120^o$
  4. $45^o$
Reveal answer Fill a bubble to check yourself
C Correct answer
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  $120$
Multiple choice mathematics and statistics hyperbola asymptote asymptotes of a curve introduction to asymptotes
Evaluate the following definite integral:
$\displaystyle \int _{0}^1 \dfrac {1-x}{1+x}dx$
  1. $2\log 2+1$
  2. $2\log 2$
  3. $2\log 2-1$
  4. $\log 4$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $

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

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

$=2\log 2-1$

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

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

  1. $(7,6)$
  2. $(1,-2)$
  3. $(10,10)$
  4. $(-5,-10)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The center of a hyperbola is the intersection of its asymptotes. Given the line passes through (-2, -6) and (4, 2), and one asymptote passes through (1, -2), one can solve for the center using the property that the midpoint of the chord is related to the center.

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

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

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a hyperbola with asymptotes L1=0 and L2=0, the equation is L1*L2 = constant. The product of perpendicular distances from any point on the hyperbola to the asymptotes is given by |constant| / sqrt(m1^2+1)*sqrt(m2^2+1). Here, the constant is 60 and the asymptotes are x+y-1=0 and x-y+3=0. The product is 60 / (sqrt(2)*sqrt(2)) = 60/2 = 30, which is divisible by 2, 3, and 5.