Parabola
Class 11th Mathematics RS Aggarwal Solution
- y2 = 12x Find the coordinates of the focus and the vertex, the equations of…
- y2 = 10x Find the coordinates of the focus and the vertex, the equations of…
- 3y2 = 8x Find the coordinates of the focus and the vertex, the equations of…
- y2 = -8x Find the coordinates of the focus and the vertex, the equations of…
- y2 = -6x Find the coordinates of the focus and the vertex, the equations of…
- 5y2 = -16x Find the coordinates of the focus and the vertex, the equations of…
- x2 = 16y Find the coordinates of the focus and the vertex, the equations of…
- x2 = 10y Find the coordinates of the focus and the vertex, the equations of…
- 3x2 = 8y Find the coordinates of the focus and the vertex, the equations of…
- x2 = -8y Find the coordinates of the focus and the vertex, the equations of…
- x2 = -18y Find the coordinates of the focus and the vertex, the equations of…
- 3x2 = -16y Find the coordinates of the focus and the vertex, the equations of…
- Find the equation of the parabola with vertex at the origin and focus at F(-2,…
- Find the equation of the parabola with focus F(4, 0) and directrix x = -4.…
- Find the equation of the parabola with focus F(0, -3) and directrix y = 3.…
- Find the equation of the parabola with vertex at the origin and focus F(0, 5).…
- Find the equation of the parabola with vertex at the origin, passing through…
- Find the equation of the parabola, which is symmetric about the y-axis and…
Exercise 22
Question 1.Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
y2 = 12x
Answer:
Given equation : y2 = 12x
Comparing given equation with parabola having equation,
y2 = 4ax
4a = 12
• a =3
Focus : F(a,0) = F(3,0)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x+a=0
• x+3=0
• x = -3
Lenth of latusrectum : 4a = 4.(3) = 12
Question 2.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
y2 = 10x
Answer:
Given equation : y2 = 10x
Comparing given equation with parabola having equation,
y2 = 4ax
4a = 10
• a =2.5
Focus : F(a,0) = F(2.5,0)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x+a=0
• x+2.5=0
• x = -2.5
Lenth of latusrectum : 4a = 4.(2.5) = 10
Question 3.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
3y2 = 8x
Answer:
Given equation :
3y2 = 8x
• 
Comparing the given equation with parabola having equation,
y2 = 4ax
• 
Focus : F(a,0) = 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x+a=0
• 
• 
Lenth of latusrectum :
Question 4.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
y2 = -8x
Answer:
Given equation :
y2 = -8x
Comparing given equation with parabola having equation,
y2 = - 4ax
4a = 8
• a = 2
Focus : F(-a,0) = F(-2,0)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x – a = 0
• x – 2 = 0
• x = 2
Lenth of latusrectum : 4a = 8
Question 5.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
y2 = -6x
Answer:
Given equation :
y2 = -6x
Comparing given equation with parabola having equation,
y2 = - 4ax
4a = 6
• 
Focus : F(-a,0) 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x – a = 0
• 
• 
Lenth of latusrectum : 4a = 6
Question 6.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
5y2 = -16x
Answer:
Given equation :
5y2 = -16x
• 
Comparing the given equation with parabola having an equation,
y2 = - 4ax
• 
• 
Focus : F(-a,0) 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : x – a = 0
• 
• 
Lenth of latusrectum :
Question 7.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
x2 = 16y
Answer:
Given equation : x2 = 16y
Comparing given equation with parabola having equation,
x2 = 4ay
4a = 16
• a = 4
Focus : F(0,a) = F(0,4)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y+a=0
• y + 4=0
• y = -4
Lenth of latusrectum : 4a = 16
Question 8.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
x2 = 10y
Answer:
Given equation : x2 = 10y
Comparing given equation with parabola having equation,
x2 = 4ay
4a = 10
• a = 2.5
Focus : F(0,a) = F(0,2.5)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y+a=0
• y + 2.5=0
• y = -2.5
Lenth of latusrectum : 4a = 10
Question 9.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
3x2 = 8y
Answer:
Given equation :
3x2 = 8y
• 
Comparing the given equation with parabola having an equation,
x2 = 4ay
• 
• 
Focus : F(0,a) 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y + a = 0
• 
• 
Lenth of latusrectum :
Question 10.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
x2 = -8y
Answer:
Given equation : x2 = - 8y
Comparing given equation with parabola having equation,
x2 = - 4ay
4a = 8
• a = 2
Focus : F(0,-a) = F(0,-2)
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y - a=0
• y - 2=0
• y = 2
Lenth of latusrectum : 4a = 8
Question 11.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
x2 = -18y
Answer:
Given equation : x2 = - 18y
Comparing given equation with parabola having equation,
x2 = - 4ay
4a = 18
• 
Focus : F(0,-a) 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y - a=0
• 
• 
Lenth of latusrectum : 4a = 18
Question 12.
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
3x2 = -16y
Answer:
Given equation :
3x2 = -16y
• 
Comparing the given equation with parabola having an equation,
x2 = 4ay
• 
• 
Focus : F(0,-a) 
Vertex : A(0,0) = A(0,0)
Equation of the directrix : y - a = 0
• 
• 
Lenth of latusrectum :
Question 13.
Find the equation of the parabola with vertex at the origin and focus at F(-2, 0).
Answer:
Vertex : A (0,0)
Given focus F(-2,0) is of the form F(-a,0)
For Vertex A(0,0) and Focus F(-a,0), equation of parabola is
y2 = - 4ax
Here, a = 2
Therefore, equation of parabola,
y2 = - 8x
Question 14.
Find the equation of the parabola with focus F(4, 0) and directrix x = -4.
Answer:
Given equation of directrix : x = -4
• x + 4 = 0
Above equation is of the form, x + a = 0
Focus of the parabola F(4,0) is of the form F(a,0)
Therefore, a = 4
For directrix with equation x+a=0 and focus (a,0), equation of the parabola is,
y2 = 4ax
• y2 = 16x
Question 15.
Find the equation of the parabola with focus F(0, -3) and directrix y = 3.
Answer:
Given equation of directrix : y = 3
• y - 3 = 0
Above equation is of the form, y - a = 0
Focus of the parabola F(0,-3) is of the form F(0,-a)
Therefore, a = 3
For directrix with equation y-a=0 and focus (0,-a), equation of the parabola is,
x2 = - 4ay
• x2 = - 12y
Question 16.
Find the equation of the parabola with vertex at the origin and focus F(0, 5).
Answer:
Vertex : A (0,0)
Given focus F(0,5) is of the form F(0,a)
For Vertex A(0,0) and Focus F(0,a), equation of parabola is
x2 = 4ay
Here, a = 5
Therefore, equation of parabola,
x2 = 20y
Question 17.
Find the equation of the parabola with vertex at the origin, passing through the point P(5, 2) and symmetric with respect to the y-axis.
Answer:
The equation of a parabola with vertex at the origin and symmetric about the y-axis is
x2 = 4ay
Since point P(5,2) passes through above parabola we can write,
52 = 4a(2)
• 25 = 8a
• 
Therefore, the equation of a parabola is
• 
• 
• 2x2 = 25y
Question 18.
Find the equation of the parabola, which is symmetric about the y-axis and passes through the point P(2, -3).
Answer:
The equation of a parabola with vertex at the origin and symmetric about the y-axis is
x2 = 4ay
Since point P(2,-3) passes through above parabola we can write,
22 = 4a(-3)
• 4 = -12a
• 
Therefore, the equation of a parabola is
• 
• 
• 3x2 = -4y
