Tag: mathematics and statistics

Questions Related to mathematics and statistics

The vertices and the foci of a hyperbola are the points $\displaystyle \left ( \pm 5, 0 \right )$ and $\displaystyle \left ( \pm 7, 0 \right )$.Which of the following holds true?

mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

  1. $\displaystyle a^{2}\neq b^{2}$

  2. $a^2=b^2$

  3. $\dfrac{a^2}{b^2}=2$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\displaystyle a= 5, ae = 7\Rightarrow a^{2}e^{2}= 49$
or $\displaystyle a^{2}\left ( 1+\frac{b^{2}}{a^{2}} \right )= 49$ 

Or 
$\displaystyle a^{2}+b^{2}=49$ 
$\displaystyle b^{2}= 24$ 
Since, $\displaystyle a^{2}\neq b^{2},$ hence hyperbola is not rectangular.

An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then

mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

  1. equation of ellipse is $x^{2}+2y^{2}=2$

  2. the foci of ellipse are $\left ( \pm 1, 0 \right )$

  3. equation of ellipse is $x^{2}+2y^{2}=4$

  4. the foci of ellipse are $\left ( \pm \sqrt{2}, 0 \right )$

Show answer
Correct Option: A,B
Explanation:

Eccentricity of the hyperbola is $\sqrt{2}$ as it is a rectangular hyperbola. 

So eccentricity $e$ of the ellipse is $\dfrac1{\sqrt{2}}$
Let the equation of the ellipse be $\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $b^{2}=a^{2}\left ( 1-e^{2} \right )=\dfrac{a^{2}}2\Rightarrow a^{2}=2b^{2}$
So equation of the ellipse is $x^{2}+2y^{2}=a^{2}$
Let $\left ( x _{1}, y _{1} \right )$ be a point of intersection of the ellipse and the hyperbola
Then $2x _{1}^{2}-2y _{1}^{2}=1$ and $x _{1}^{2}+2y _{1}^{2}=a^{2}$          (1)
Equations of the tangents at $\left ( x _{1}, y _{1} \right )$ to the two conics are
   $2xx _{1}-2yy _{1}=1$ and $xx _{1}+2yy _{1}=a^{2}$
Since the two conics intersect orthogonally
$\displaystyle \left ( \frac{x _{1}}{y _{1}} \right )\left ( -\frac{x _{1}}{2y _{1}} \right )=-1\Rightarrow x _{1}^{2}=2y _{1}^{2}$
And from (1) we get $x _{1}^{2}=1$, $a^{2}=2$.
Hence the equation of the ellipse is $x^{2}+2y^{2}=2$ and its focus is
   $\displaystyle \left ( \pm ae, 0 \right )=\left ( \pm \sqrt{2}\times \frac{1}{\sqrt{2}}, 0 \right )=\left ( \pm 1, 0 \right )$

The equation of the ellipse whose equation of directrix is $3x+4y-5=0$, coordinates of the focus are $(1,2)$ and the eccentricity is $\dfrac{1}{2}$ is $91x^2+84y^2-24xy-170x-360y+475=0$

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Let $P(x,y)$ be any point on the ellipse and PM be the perpendicular from P upon the directrix $3x+4y-5=0$.

Then by the definition,
$\dfrac{SP}{PM}=e$

$SP=e.PM$
$\sqrt{(x-1)^2+(y-2)^2}=\dfrac{1}{2}|\dfrac{3x+4y-5}{\sqrt{3^2+4^2}}|$

$(x-1)^2+(y-2)^2=\dfrac{1}{4}. \dfrac{(3x+4y-5)^2}{25}$

$100(x^2+y^2-2x-4y+5)=9x^2+16y^2+24xy-30x-40y+25$
$91x^2+84y^2-24xy-170x-360y+475=0$ is the equation of the ellipse.

The equation of the ellipse whose foci are $(\pm5,0)$ and of the directrix is $5x=36$, is

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $\dfrac{x^2}{36}+\dfrac{y^2}{11}=1$

  2. $\dfrac{x^2}{6}+\dfrac{y^2}{\sqrt{11}}=1$

  3. $\dfrac{x^2}{6}+\dfrac{y^2}{11}=1$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given foci $(\pm 5,0)$ and directrix $x=\cfrac{36}{5}$

Then $ae=5$ (focus coordinates ($\pm ae,0)]$....(1)
$\cfrac{a}{e}=\cfrac{36}{5}$ (directrix equation $x=\cfrac{a}{e}$]....(2)
From (1) and (2) ${a}^{2}=36\Rightarrow$ $a=6$
$e=\cfrac{5}{6}\Rightarrow $ $\sqrt { 1-\cfrac { { b }^{ 2 } }{ { a }^{ 2 } }  } =\cfrac { 5 }{ 6 } $
$1-\cfrac { { b }^{ 2 } }{ 36 } =\cfrac{25}{36}$
$b=\sqrt 11$
required equation $\cfrac{{x}^{2}}{36}+\cfrac{{y}^{2}}{11}=1$

If the eccentricity of the ellipse $\dfrac{x^2}{a^2 + 1} + \dfrac{y^2}{a^2 + 2 } = 1$ is $\dfrac{1}{\sqrt{6}}$, then the length of latusrectum is

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $\dfrac{5}{\sqrt{6}}$

  2. $\dfrac{10}{\sqrt{6}}$

  3. $\dfrac{8}{\sqrt{6}}$

  4. None of these

Show answer
Correct Option: B

If focus of the parabola is $(3,0)$ and length of latus rectum is $8$, then its vertex is

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $(2,0)$

  2. $(1,0)$

  3. $(0,0)$

  4. $(-1,0)$

Show answer
Correct Option: B
Explanation:
Given, focus $=(3,0)$ and Length of latus rectum $= 8$

$\Rightarrow 4a=8$ $\Rightarrow a=2$

$\Rightarrow$ Vertex = $(3-a,0)$ $=(1,0)$

$\therefore $ Option B is correct

If $(0,0)$ be the vertex and $3x-4y+2=0$ be the directrix of a parabola, then the length of its latus rectum is

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $4/5$

  2. $2/5$

  3. $8/5$

  4. $1/5$

Show answer
Correct Option: C
Explanation:
Distance of vertex from directrix = $\dfrac{\left | 3(0)-4(0)+2 \right |}{\sqrt{3^{2}+4^{2}}}= \dfrac{2}{5}=a$

Length of latus rectum = $4a= \dfrac{8}{5}$

$\therefore $ Option C is correct

Eccentricity of an ellipse is $\sqrt {\cfrac{2}{5}} $ and it passes through the point $(-3,1)$ then its equation is 

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $3{x^2} + 5{y^2} = 32$

  2. $2{x^2} + 3{y^2} = 33$

  3. $3{x^2} + 4{y^2} = 30$

  4. $2{x^2} + 3{y^2} = 34$

Show answer
Correct Option: B

If $P = (x, y), F _1 = (3, 0)$ and $16x^2 + 25y^2 = 400$, then $PF _1 + PF _2$ equals

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $8$

  2. $6$

  3. $10$

  4. $12$

Show answer
Correct Option: C
Explanation:
$PF _1+PF _2=2a$

$\dfrac{x^2}{5^2}+\dfrac{y^2}{4^2}=1\Rightarrow a=5,b=4$

$\therefore PF _1+PF _2=2(5)$

$=10$

Which of the following can be the equation of an ellipse?

mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $x^{2} + y^{2} = 5$

  2. $\dfrac {x^{2}}{9} + \dfrac {x^{2}}{9} = 1$

  3. $2x^{2} + 3y^{2} = 5$

  4. $2x + 2y = 5$

Show answer
Correct Option: C