RD Sharma Solutions for Class 11 Maths Chapter 25 Parabola

RD Sharma Solutions Class 11 Maths Chapter 25 – Get Free PDF Updated for 2023-24

RD Sharma Solutions for Class 11 Maths Chapter 25 – Parabola is provided here for students to enhance their learning process and score good marks in the board exams. In this section, we shall study the parabola and also find the general equation of a conic section when its focus, directrix and eccentricity are given. For students who have difficulty understanding the concepts during class hours, the subject experts at BYJU’S have prepared the RD Sharma Solutions, keeping in mind the students’ grasping abilities. To boost interest among students in this chapter, RD Sharma Class 11 Maths Solutions Chapter 25 Parabola PDF links are given here for easy access.

Chapter 25 – Parabola contains one exercise, and RD Sharma Solutions offers accurate answers to each question present in this exercise. Students can also access online and can solve both chapter-wise and exercise-wise problems to increase their confidence level before appearing for the board exam.  Now, let us have a look at the concepts discussed in this chapter.

• Conic sections.
• Analytical definition of conic section.
• General equation of a conic section when its focus, directrix and eccentricity are given.
• The parabola.
• Some applications of parabola.

RD Sharma Solutions for Class 11 Maths Chapter 25 – Parabola

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Access RD Sharma Solutions for Class 11 Maths Chapter 25 – Parabola

EXERCISE 25.1 PAGE NO: 25.24

1. Find the equation of the parabola whose:
(i) focus is (3, 0) and the directrix is 3x + 4y = 1

(ii) focus is (1, 1) and the directrix is x + y + 1 = 0

(iii) focus is (0, 0) and the directrix is 2x – y – 1 = 0

(iv) focus is (2, 3) and the directrix is x – 4y + 1 = 0

Solution:

(i) The focus is (3, 0) and the directrix is 3x + 4y = 1

Given:

The focus S(3, 0) and directrix(M) 3x + 4y – 1 = 0.

Let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Upon cross-multiplication, we get

25x² + 25y² – 150x + 225 = 9x² + 16y² – 6x – 8y + 24xy + 1

16x² + 9y² – 24xy – 144x + 8y + 224 = 0

∴The equation of the parabola is 16x² + 9y² – 24xy – 144x + 8y + 224 = 0

(ii) The focus is (1, 1) and the directrix is x + y + 1 = 0

Given:

The focus S(1, 1) and directrix(M) x + y + 1 = 0.

Let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Upon cross-multiplication, we get

2x² + 2y² – 4x – 4y + 4 = x² + y² + 2x + 2y + 2xy + 1

x² + y² + 2xy – 6x – 6y + 3 = 0

∴ The equation of the parabola is x² + y² + 2xy – 6x – 6y + 3 = 0

(iii) The focus is (0, 0) and the directrix is 2x – y – 1 = 0

Given:

The focus S(0, 0) and directrix(M) 2x – y – 1 = 0.

Let us assume P(x, y) be any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Upon cross-multiplication, we get

5x² + 5y² = 4x² + y² – 4x + 2y – 4xy + 1

x² + 4y² + 4xy + 4x – 2y – 1 = 0

∴ The equation of the parabola is x² + 4y² + 4xy + 4x – 2y – 1 = 0

(iv) The focus is (2, 3) and the directrix is x – 4y + 1 = 0

Given:

The focus S(2, 3) and directrix(M) x – 4y + 3 = 0.

Let us assume P(x, y) be any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Upon cross-multiplication, we get

17x² + 17y² – 68x – 102y + 221 = x² + 16y² + 6x – 24y – 8xy + 9

16x² + y² + 8xy – 74x – 78y + 212 = 0

∴ The equation of the parabola is 16x² + y² + 8xy – 74x – 78y + 212 = 0

2. Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0. Also, find the length of its latus – rectum.

Solution:

Given:

The focus S(2, 3) and directrix(M) x – 4y + 3 = 0.

Let us assume P(x, y) be any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Upon cross-multiplication, we get

17x² + 17y² – 68x – 102y + 221 = x² + 16y² + 6x – 24y – 8xy + 9

16x² + y² + 8xy – 74x – 78y + 212 = 0

∴ The equation of the parabola is 16x² + y² + 8xy – 74x – 78y + 212 = 0.

Now, let us find the length of the latus rectum,

We know that the length of the latus rectum is twice the perpendicular distance from the focus to the directrix.

So by using the formula,

∴ The length of the latus rectum is 14/√17

3. Find the equation of the parabola, if
(i) the focus is at (-6, 6) and the vertex is at (-2, 2)

(ii) the focus is at (0, -3) and the vertex is at (0, 0)

(iii) the focus is at (0, -3) and the vertex is at (-1, -3)

(iv) the focus is at (a, 0) and the vertex is at (a′, 0)

(v) the focus is at (0, 0) and the vertex is at the intersection of the lines x + y = 1 and x – y = 3.

Solution:

(i) The focus is at (-6, 6) and the vertex is at (-2, 2)

Given:

Focus = (-6, 6)

Vertex = (-2, 2)

Let us find the slope of the axis (m₁) = (6-2)/(-6-(-2))

= 4/-4

= -1

Let us assume m₂ be the slope of the directrix.

m₁m₂ = -1

-1m₂ = -1

m₂ = 1

Now, let us find the point on directrix.

(-2, 2) = [(x-6/2), (y+6)/2]

By equating, we get,

(x-6/2) = -2 and (y+6)/2 = 2

x-6 = -4 and y+6 = 4

x = -4+6 and y = 4-6

x = 2 and y = -2

So the point of directrix is (2, -2).

We know that the equation of the lines passing through (x₁, y₁) and having slope m is y – y₁ = m(x – x₁)

y – (-2) = 1(x – 2)

y + 2 = x – 2

x – y – 4 = 0

Let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Now by cross-multiplying, we get

2x² + 2y² + 24x – 24y + 144 = x² + y² – 8x + 8y – 2xy + 16

x² + y² + 2xy + 32x – 32y + 128 = 0

∴ The equation of the parabola is x² + y² + 2xy + 32x – 32y + 128 = 0

(ii) The focus is at (0, -3) and the vertex is at (0, 0)

Given:

Focus = (0, -3)

Vertex = (0, 0)

Let us find the slope of the axis (m₁) = (-3-0)/(0-0)

= -3/0

= ∞

Since the axis is parallel to the x-axis, the slope of the directrix is equal to the slope of the x-axis = 0

So, m₂ = 0

Now, let us find the point on directrix.

(0, 0) = [(x-0/2), (y-3)/2]

By equating, we get,

(x/2) = 0 and (y-3)/2 = 0

x = 0 and y – 3 = 0

x = 0 and y = 3

So the point on directrix is (0, 3).

We know that the equation of the lines passing through (x₁, y₁) and having slope m is y – y₁ = m(x – x₁)

y – 3 = 0(x – 0)

y – 3 = 0

Now, let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

Now by cross-multiplying, we get

x² + y² + 6y + 9 = y² – 6y + 9

x² + 12y = 0

∴ The equation of the parabola is x² + 12y = 0

(iii) The focus is at (0, -3) and the vertex is at (-1, -3)

Given:

Focus = (0, -3)

Vertex = (-1, -3)

Let us find the slope of the axis (m₁) = (-3-(-3))/(0-(-1))

= 0/1

= 0

We know the product of the slopes of the perpendicular lines is – 1 for non – vertical lines.

Here the slope of the axis is equal to the slope of the x-axis. So, the slope of directrix is equal to the slope of the y-axis, i.e., ∞.

So, m₂ = ∞

Now let us find the point on directrix.

(-1, -3) = [(x+0/2), (y-3)/2]

By equating, we get,

(x/2) = -1 and (y-3)/2 = -3

x = -2 and y – 3 = -6

x = -2 and y = -6+3

x = -2 and y = -3

So, the point on directrix is (-2, -3)

We know that the equation of the lines passing through (x₁, y₁) and having slope m is y – y₁ = m(x – x₁)

y – (- 3) = ∞(x – (- 2))

(y+3)/ ∞ = x + 2

x + 2 = 0

Now, let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

By cross-multiplying, we get

x² + y² + 6y + 9 = x² + 4x + 4

y² – 4x + 6y + 5 = 0

∴ The equation of the parabola is y² – 4x + 6y + 5 = 0

(iv) the focus is at (a, 0) and the vertex is at (a′, 0)

Given:

Focus = (a, 0)

Vertex = (a′, 0)

Let us find the slope of the axis (m₁) = (0-0)/(a′, a)

= 0/(a′, a)

= 0

We know the product of the slopes of the perpendicular lines is – 1 for non-vertical lines.

Here the slope of the axis is equal to the slope of the x-axis. So, the slope of directrix is equal to the slope of the y-axis, i.e., ∞.

So, m₂ = ∞

Now let us find the point on directrix.

(a′, 0)= [(x+a/2), (y+0)/2]

By equating, we get,

(x+a/2) = a′ and (y)/2 = 0

x + a = 2a′ and y = 0

x = (2a′ – a) and y = 0

So the point on directrix is (2a′ – a, 0).

We know that the equation of the lines passing through (x₁, y₁) and having slope m is y – y₁ = m(x – x₁)

y – (0) = ∞(x – (2a′ – a))

y/∞ = x + a – 2a′

x + a – 2a′ = 0

Now, let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

By cross-multiplying, we get,

x² + y² – 2ax + a² = x² + a² + 4(a′)² + 2ax – 4aa′ – 4a′x

y² – (4a – 4a′)x + a² – 4(a′)² + 4aa′ = 0

∴ The equation of the parabola is y² – (4a – 4a′)x + a² – 4(a′)² + 4aa′ = 0

(v) the focus is at (0, 0), and the vertex is at the intersection of the lines x + y = 1 and x – y = 3.

Given:

Focus = (0, 0)

Vertex = intersection of the lines x + y = 1 and x – y = 3.

So the intersecting point of the above lines is (2, -1)

Vertex = (2, -1)

Slope of axis (m₁) = (-1-0)/(2-0)

= -1/2

We know that the product of the slopes of the perpendicular lines is – 1.

Let us assume m₂ be the slope of the directrix.

m₁.m₂ = -1

-1/2 . m₂ = -1

So m₂ = 2

Now let us find the point on directrix.

(2, -1) = [(x+0)/2, (y+0)/2]

(x)/2 = 2 and y/2 = -1

x = 4 and y = -2

The point on directrix is (4, – 2).

We know that the equation of the lines passing through (x₁, y₁) and having slope m is y – y₁ = m(x – x₁)

y – (- 2) = 2(x – 4)

y + 2 = 2x – 8

2x – y – 10 = 0

Now, let us assume P(x, y) is any point on the parabola.

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is

So by equating both, we get

By cross-multiplying, we get

5x² + 5y² = 4x² + y² – 40x + 20y – 4xy + 100

x² + 4y² + 4xy + 40x – 20y – 100 = 0

∴ The equation of the parabola is x² + 4y² + 4xy + 40x – 20y – 100 = 0

4. Find the vertex, focus, axis, directrix and lotus – rectum of the following parabolas
(i) y² = 8x

(ii) 4x² + y = 0

(iii) y² – 4y – 3x + 1 = 0

(iv) y² – 4y + 4x = 0

(v) y² + 4x + 4y – 3 = 0

Solution:

(i) y² = 8x

Given:

Parabola = y² = 8x

Now by comparing with the actual parabola y² = 4ax

Then,

4a = 8

a = 8/4 = 2

So, the vertex is (0, 0)

The focus is (a, 0) = (2, 0)

The equation of the axis is y = 0.

The equation of the directrix is x = – a i.e., x = – 2

The length of the latus rectum is 4a = 8.

(ii) 4x² + y = 0

Given:

Parabola => 4x² + y = 0

Now by comparing with the actual parabola y² = 4ax

Then,

4a = ¼

a = 1/(4 × 4)

= 1/16

So, the vertex is (0, 0)

The focus is = (0, -1/16)

The equation of the axis is x = 0.

The equation of the directrix is y = 1/16

The length of the latus rectum is 4a = ¼

(iii) y² – 4y – 3x + 1 = 0

Given:

Parabola y² – 4y – 3x + 1 = 0

y² – 4y = 3x – 1

y² – 4y + 4 = 3x + 3

(y – 2)² = 3(x + 1)

Now by comparing with the actual parabola y² = 4ax

Then,

4b = 3

b = ¾

So, the vertex is (-1, 2)

The focus is = (3/4 – 1, 2) = (-1/4, 2)

The equation of the axis is y – 2 = 0.

The equation of the directrix is (x – c) = -b

(x – (-1)) = -3/4

x = -1 – ¾

= -7/4

The length of the latus rectum is 4b = 3

(iv) y² – 4y + 4x = 0

Given:

Parabola y² – 4y + 4x = 0

y² – 4y = – 4x

y² – 4y + 4 = – 4x + 4

(y – 2)² = – 4(x – 1)

Now by comparing with the actual parabola y² = 4ax => (y – a)² = – 4b(x – c)

Then,

4b = 4

b = 1

So, the vertex is (c, a) = (1, 2)

The focus is (b + c, a) = (1-1, 2) = (0, 2)

The equation of the axis is y – a = 0 i.e., y – 2 = 0

The equation of the directrix is x – c = b

x – 1 = 1

x = 1 + 1

= 2

The length of latus rectum is 4b = 4

(v) y² + 4x + 4y – 3 = 0

Given:

The parabola y² + 4x + 4y – 3 = 0

y² + 4y = – 4x + 3

y² + 4y + 4 = – 4x + 7

(y + 2)² = – 4(x – 7/4)

Now by comparing with the actual parabola y² = 4ax => (y – a)² = – 4b(x – c)

Then,

4b = 4

b = 4/4 = 1

So, The vertex is (c, a) = (7/4, -2)

The focus is (- b + c, a) = (-1 + 7/4, -2) = (3/4, -2)

The equation of the axis is y – a = 0 i.e., y + 2 = 0

The equation of the directrix is x – c = b

x – 7/4 = 1

x = 1 + 7/4

= 11/4

The length of latus rectum is 4b = 4.

5. For the parabola, y² = 4px find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

Solution:

Given:

The parabola, y² = 4px and a double ordinate of length 8p.

Let AB be the double ordinate of length 8p for the parabola y² = 4px.

Now, let us compare to the actual parabola, y² = 4ax

Then,

axis is y = 0

vertex is O(0, 0).

We know that double ordinate is perpendicular to the axis.

So, let us assume that the point at which the double ordinate meets the axis is (x₁, 0).

Then the equation of the double ordinate is y = x₁. It meets the parabola at the points (x₁, 4p) and (x₁, – 4p) as its length is 8p.

Now, let us find the value of x₁ by substituting in the parabola.

(4p)² = 4p(x₁)

x₁ = 4p.

The extremities of the double ordinate are A(4p, 4p) and B(4p, – 4p).

Assume the slopes of OA and OB be m₁ and m₂. Let us find their values.

m₁ = (4p – 0)/(4p – 0)

= 4p/4p

= 1

m₂ = (4p – 0)/(-4p – 0)

= 4p/-4p

= -1

So, m₁.m₂ = 1. – 1

= – 1

The product of slopes is -1. So, the lines OA and OB are perpendicular to each other.

Hence the extremities of double ordinate make a right angle with the vertex.

6. Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus – rectum.

Solution:

Given:

The parabola, x² = 12y

Now, let us compare to the actual parabola, y² = 4ax

Then,

Vertex is O(0, 0)

Ends of latus rectum is (2b, b), (-2b, b)

4b = 12

b = 12/4

= 3

Ends of latus rectum = (2(3), 3), (- 2(3), 3)

Ends of latus rectum is A(6, 3), B(- 6, 3)

We know that the area of the triangle with the vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

∴The area of the triangle is 18 sq. units.

7. Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is (3, 3) and directrix is 3x – 4y = 2. Find also the length of the latus-rectum.

Solution:

Given:

Focus = (3, 3)

Directrix = 3x – 4y = 2

Firstly let us find the slope of the directrix.

The slope of the line ax + by + c = 0 is –a/b

So, m₁ = -3/-4

= ¾

Let us assume the slope of the axis is m₂.

m₁.m₂ = -1

¾ . m₂ = -1

m₂ = -4/3

We know that the equation of the line passing through the point (x₁, y₁) and having slope m is (y – y₁) = m(x – x₁)

y – 3 = -4/3 (x – 3)

3(y – 3) = – 4(x – 3)

3y – 9 = – 4x + 12

4x + 3y = 21

On solving the lines, the intersection point is (18/5, 11/5)

By using the formula to find the length is given as

∴The length of the latus rectum is 2.

8. At what point of the parabola x² = 9y is the abscissa three times that of ordinate?

Solution:

Given:

The parabola, x² = 9y

Let us assume the point is (3y₁, y₁).

Now by substituting the point in the parabola, we get,

(3y₁)² = 9(y₁)

9y₁² = 9y₁

y₁² – y₁ = 0

y₁(y₁ – 1) = 0

y₁ = 0 or y₁ – 1 = 0

y₁ = 0 or y₁ = 1

The points is B (3(1), 1) => (3, 1)

∴The point is (3, 1).