Tag: mathematics and statistics

Questions Related to mathematics and statistics

The focus of the parabola $y ^ { 2 } = 4 y - 4 x$ is

  1. $( 0,2 )$

  2. $( 1,2 )$

  3. $( 4,2 )$

  4. $( 1,3 )$


Correct Option: A
Explanation:

$\Rightarrow y^2 = 4y - 4x$

$\Rightarrow y^2 - 4y = -4x$
$\Rightarrow y^2 - 4y + 4 = -4x + y$
$\Rightarrow (y - 2)^2 = -4 (x - 1)$
vertex = $(1, 2)$
$\dfrac{1}{4 p} = \dfrac{-1}{4} \rightarrow p = -1$
$\therefore$ Focus = $(-1 + 1, 2 + 0) = (0, 2)$

Equation of the directrix of the parabola whose focus is $(0,0)$ and the tangent at the vertex is $x-y+1=0$ is 

  1. $x-y=0$

  2. $x-y-1=0$

  3. $x-y+2=0$

  4. $x+y-1=0$


Correct Option: A
Explanation:

Putting $x=1,y=2$

$1-2+1=0\-1+1=0$
So, equation of directrix of parabola $x-y=0$

The equation of the directrix of the parabola, $y ^ { 2 } + 4 y + 4 x + 2 = 0$ is -

  1. $x = - 1$

  2. $x = 1$

  3. $x = - \dfrac {3 }{ 2}$

  4. $x = \dfrac {3 }{ 2}$


Correct Option: D
Explanation:
Given equation of parabola
$y^2+4y+4x+2=0$
or $(y+2)^2+4x+2=0$
or $(y+2)^2=-4(x-12)$
Its directrix  is x12x−12=1=1
or x=32
Hence D is the correct answer.

y2+4y+4x+2=

The equation of directrix of the parabola $(y-2)^{2}=4(x-4)$, is

  1. $x+1=0$

  2. $x=1$

  3. $x=2$

  4. $x=4$


Correct Option: A

The vertex of the parabola $ {4y}^{2} + 12x-12y+39= 0$ is:

  1. $ \left( 5/2,3/2 \right)$

  2. $ \left ( -5/2,3/2 \right) $

  3. $ \left ( -5/2,-3/2 \right )$

  4. $ \left ( 5/2,-3/2 \right )$


Correct Option: A

The focus of the parabola $(y-2)^{2}=20(x+3)$ is:

  1. $(2, 2)$

  2. $(0, 2)$

  3. $(3, -2)$

  4. $(-2, 0)$


Correct Option: A

A parabola is written as $x^{2}=4ay$, its focus and equation of the directrix is:

  1. $(a, 0)$ and $x+a=0$

  2. $(0, a)$ and $y+a=0$

  3. $(0, a)$ and $x-a=0$

  4. $(0, a)$ and $y-a=0$


Correct Option: A

Focus of the parabola $4x^{2}-12x+8y+13=0$ is

  1. $\left(\dfrac {3}{2},\ -2 \right)$

  2. $\left(\dfrac {3}{2},\ -5 \right)$

  3. $\left(\dfrac {3}{2},\ -3 \right)$

  4. $\left(\dfrac {3}{2},\ -1 \right)$


Correct Option: A

The locus of the foot of the perpendicular from the focus upon a tangent to the parabola $y^{2}=4ax$ is 

  1. $the\ directrix$

  2. $tangent\ at\ the\ vertex$

  3. $x=a$

  4. $none\ of\ these$


Correct Option: A

Maximum radius of the circle inscribed in parabola ${y}^{2}=4x$ with centre its focus is ..

  1. $8$

  2. $4$

  3. $2$

  4. $5$


Correct Option: A