CBSE Solutions for Class 11 English

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Sets Relation and Function Trignometric Functions Mathematical Induction Complex Numbers and Quardratic Equations Linear Equations Permutation and Combination Binomial Theorem Sequence and Series Straight Lines Conic Section Introduction to 3D Geometry Limits and Derivatives Mathematical Reasoning Statistics Probability
The equations of the tangents to the ellipse 9x2 + 16y2 = 144 from the point (2, 3) are ___________

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જવાબ : x + y = 5, y = 3


The latus-rectum of the conic 3x2 + 4y2 − 6x + 8y − 5 = 0 is ___________

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જવાબ : 3


The eccentricity of the conic 9x2 + 25y2 = 225 is ___________

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જવાબ : 4/5


The difference between the lengths of the major axis and the latus-rectum of an ellipse is ___________

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જવાબ : 2ae2


The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is ___________

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જવાબ : 1/√2


The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is ___________

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જવાબ : √5-1/2


The eccentricity of the ellipse x2/a2 + y2/b2 =1 if its latus rectum is equal to one half of its minor axis, is___________

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જવાબ : √3/2


The equation of the circle drawn with the two foci of x2/a2 + y2/b2 = 1 as the end-points of a diameter is___________

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જવાબ : x2 + y2 = a2 − b2


If the equation of a circle is 2λx2 + (4λ − 6)y2 − 8x + 12y − 2 = 0, then the coordinates of centre are _________

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જવાબ : (2/3, −1)


The equation x2 + y2 + 2x − 4y + 5 = 0 represents _____________

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જવાબ : a point


If the centroid of an equilateral triangle is (1, 1) and its one vertex is (2, 2), then the equation of its circumcircle is _____________

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જવાબ : x2 + y2 − 2x − 2y = 0


The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 is_____________

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જવાબ : 4 (x2 + y2 − x − y) + 1 = 0


the circles x2 + y2 = 9 and x2 + y2 + 8y + 2c = 0 touch each other, then c is equal to _____________

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જવાબ : 15


If the circle x2 + y2 + 2ax + 6y + 9 = 0 touches x-axis, then the value of a is _____________

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જવાબ : ± 3


The equation of a circle with radius 4 and touching both the coordinate axes is _____________

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જવાબ : x2 + y2 ± 8x ± 8y + 16 = 0


The equation of the circle passing through the origin which cuts off intercept of length 6 and 6 from the axes is _____________

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જવાબ : x2 + y2 − 6x − 7y = 3√2 - 18


The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if g2 _____________

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જવાબ : < c


If (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y), then the values of x and y are _____________

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જવાબ : x = 1, y = 4


Equation of the diameter of the circle x2 + y2 + 4x - 2y = 0 which passes through the origin is _____________

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જવાબ : x + 2y = 0


The vertex of the parabola (y + k)2 = 8k (x − k) is _____________

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જવાબ : (k, −k)


The equation of the parabola whose vertex is (k, 0) and the directrix has the equation x + y = 3k, is _____________

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જવાબ : x2 − 2xy + y2 + 6kx + 10ky – 7k2 = 0


The locus of the points of trisection of the double ordinates of a parabola is a _____________

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જવાબ : x2 − 2xy + y2 + 6kx + 10ky – 7k2 = 0


The equation of the parabola with focus (0, 0) and directrix x + y = 7 is _____________

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જવાબ : x2 + y2 − 2xy + 14x + 14y − 49 = 0


The equation of the ellipse with focus (−1, 1), directrix x − y + 3 = 0 and eccentricity 1/2 is ___________

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જવાબ : 7x2 + 2xy + 7y2 + 10x − 10y + 7 = 0


For the ellipse 12x2 + 4y2 + 24x − 16y + 25 = 0, centre is ___________

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જવાબ : (−1, 2)


The locus of the point of intersection of the lines √3x-y-4√3λ=0 and √3λx+λy-4√3=0 is a hyperbola of eccentricity ___________

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જવાબ : 2


The equation of the hyperbola whose centre is (6, 2) one focus is (4, 2) and of eccentricity 2 is ___________

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જવાબ : 3 (x − 6)2 − (y −2)2 = 3


In the parabola y2 = 4kx, the length of the chord passing through the vertex and inclined to the axis at π/4 is _____________

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જવાબ : 4√2k


The equation 9x2 + y2 + 6xy − 74x − 78y + 212 = 0 represents _____________

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જવાબ : a parabola


Which points lie on the parabola x2 = ay?

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જવાબ : x = at, y = at2


Which points lie on the parabola x2 = 9ay?

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જવાબ : x = 3at, y = 3at2


Which points lie on the parabola 16x2 = 16ay?

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જવાબ : x = at, y = at2


Equation of the hyperbola whose vertices are (± 5, 0) and foci at (± 13, 0), is_____________

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જવાબ : 144x2 − 25y2 = 3600


If e1 and e2 are respectively the eccentricities of the ellipse x2/18 + y2/4 = 1 and the hyperbola x2/9 - y2/4 = 1, then the relation between e1 and e2 is_____________

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જવાબ : 2 e12 + e22 = 3


The distance between the directrices of the hyperbola x = 8 sec θ, y = 8 tan θ, is_____________

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જવાબ : 8√2


The equation of the conic with focus at (1, −1) directrix along x − y + 1 = 0 and eccentricity √2 is_____________

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જવાબ : 2xy − 4x + 4y + 1 = 0


The eccentricity of the conic 25x2 − 144y2 = 3600 is_____________

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જવાબ : 13/12


A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA − PB = k (k ≠ 0), then the locus of P is_____________

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જવાબ : a hyperbola


The eccentricity of the hyperbola which has latus-rectum ½ of its transverse axis, is_____________

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જવાબ : √[3/2]


The eccentricity of the hyperbola x2 − 4y2 = 1 is _____________

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જવાબ : √5/2


The difference of the focal distances of any point on the hyperbola is equal to _____________

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જવાબ : length of the transverse axis


The foci of the hyperbola 25x2 − 144y2 = 3600 are _____________

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જવાબ : (± 13, 0)


The eccentricity the hyperbola x=a2(t+1t), y=a2(t-1t) is ___________

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જવાબ : √2


The foci of the hyperbola 2x2 − 3y2 = 5 are ___________

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જવાબ : (±5/√6,0)


The latus-rectum of the hyperbola 16x2 − 9y2 = 144 is ___________

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જવાબ : 32/3


The length of the straight line x − 3y = 1 intercepted by the hyperbola x2 − 4y2 = 1 is ___________

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જવાબ : 6√2/5


The distance between the foci of a hyperbola is 8 and its eccentricity is √2, then equation of the hyperbola is_____________

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જવાબ : x2 − y2 = 8


If e1 is the eccentricity of the conic 9x2 + 4y2 = 36 and e2 is the eccentricity of the conic 9x2 − 4y2 = 36, then e22 − e12 : _____________

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જવાબ : 2 < e22 − e12 < 3


If the eccentricity of the hyperbola x2 − y2 sec2α = 5 is √3 times the eccentricity of the ellipse x2 sec2 α + y2 = 25, then α =___________

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જવાબ : π/4


The equation of the hyperbola whose foci are (6, 4) and (−4, 4) and eccentricity 2, is ___________

 

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જવાબ : [(x-1)2]25/4 – [(y-4)2]75/4=1


Find the radius of each of the following circles :  x2 + y2 − 4x + 6y = 5

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
x2 + y2 − 4x + 6y = 5

The given equation can be rewritten as follows:
(x−2)2+(y+3)2−4−9=5
⇒(x−2)2 + (y+3)2 =18

Thus, the radius = √18=3√2


Find the radius of each of the following circles :   (x + 5)2 + (y + 1)2 = 9

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
(x + 5)2 + (y + 1)2 = 9
Thus, the radius = 3


Find the centre of each of the following circles :  x2 + y2 − x + 2y − 3 = 0.

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
x2 + y2− x + 2y − 3=0

The given equation can be rewritten as follows:
(x− ½ )2+(y+1)2− ¼ −1−3=0
⇒(x− ½ )2 + (y+1)2 = 17/4
Thus, the centre is ( ½ ,−1)


Find the radius of each of the following circles :  (x − 1)2 + y2 = 4

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
(x − 1)2 + y2 = 4
Thus, the radius is 2.


Find the centre of each of the following circles :  x2 + y2 − 4x + 6y = 5

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
x2 + y2 − 4x + 6y = 5

The given equation can be rewritten as follows:
(x−2)2+(y+3)2−4−9=5
⇒(x−2)2 + (y+3)2 =18

Thus, the centre is (2, −3).


Find the centre of each of the following circles :  (x + 5)2 + (y + 1)2 = 9

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
(x + 5)2 + (y + 1)2 = 9

Here, p = −5, q = −1

Thus, the centre is (-5, −1).


Find the centre of each of the following circles :  (x − 1)2 + y2 = 4

 

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2
Given:
(x − 1)2 + y2 = 4

Here, p = 1, q = 0 and a = 2

Thus, the centre is (1, 0)


Find the focus of the following parabolas  :  y2 = 8x + 8y

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જવાબ : Given:
y2 = 8x + 8y
⇒(y−4)2=8(x+2)
Putting Y=y−4, X=x+2:
Y2=8X

On comparing the given equation with Y2=4aX:
4a=8

⇒a=2

∴ Focus = (X = a, Y = 0) = (x+2=2, y−4=0)=(x=0, y=4)


Find the focus of the following parabolas  :  y2 + 4x + 4y − 3 = 0

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જવાબ : Given:
y2 + 4y + 4x −3 = 0

⇒(y+2)2− 4 + 4x – 3 = 0

⇒(y+2)2=−4 (x−7/4)
Let Y=y+2, X=x−7/4
Then, we have:
Y2=−4X

Comparing the given equation with Y2=−4aXY2=-4aX:
4a=4

⇒a=1

∴ Focus = (X = −a, Y = 0) = (x−74=−1, y+2=0)=(x=34, y=−2)


Find the focus of the following parabolas  :  y2 − 4y + 4x = 0

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જવાબ : Given:
y2 − 4y + 4x = 0

⇒(y−2)2−4+4x=0

⇒(y−2)2=−4(x−1)

Let Y=y−2, X=x−1
Then, we have:
Y2=−4X

Comparing the given equation with Y2=−4aX:

4a=4

⇒a=1

∴ Focus = (X = −a, Y = 0) = (x−1=−1, y−2=0)=(x=0, y=2)


Find the focus of the following parabolas  :  y2 − 4y − 3x + 1 = 0

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જવાબ : Given: 
y2 − 4y − 3x + 1 = 0

⇒(y−2)2−4−3x+1=0

⇒(y−2)2 = 3(x+1)

⇒(y−2)2= 3(x−(−1))
Let Y=y−2, X=x+1
Then, we have:
Y2=3X
Comparing the given equation with Y2=4aX:
4a=3⇒a= ¾
∴ Focus = (X = a, Y = 0) = (x+1=34, y−2=0)=(x=−14, y=2)


Find the focus of the following parabolas  :  4x2 + y = 0

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જવાબ : Given:
  4x2 + y = 0

⇒−y/4=x2

On comparing the given equation with x2=−4ay:

4a=1/4⇒

a=116

∴ Focus = (0, −a) = (0,−116)


Find the focus of the following parabolas  :  y2 = 8x

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જવાબ : Given:
  y2 = 8x
On comparing the given equation with y2=4ax:
4a=8

⇒a=2

∴ Focus = (a, 0) = (2, 0)


Find the vertex of the following parabolas  :  y2 = 8x + 8y

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જવાબ : Given:
y2 = 8x + 8y
⇒(y−4)2=8(x+2)

Putting Y=y−4, X=x+2:
Y2=8X

On comparing the given equation with Y2=4aX:
4a=8

⇒a=2

∴ Vertex = (X = 0, Y = 0) = (x=−2, y=4)x=-2, y=4


Find the vertex of the following parabolas  :  y2 + 4x + 4y − 3 = 0

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જવાબ : Given:
y2 + 4y + 4x −3 = 0

⇒(y+2)2−4+4x−3=0

⇒(y+2)2=−4(x−7/4)

Let Y=y+2, X=x−7/4
Then, we have:
Y2=−4XY

Comparing the given equation with Y2=−4a:
4a=4

⇒a=1

∴ Vertex = (X = 0, Y = 0) = (x=74, y=−2)x=74, y=-2


Find the vertex of the following parabolas  :  y2 − 4y + 4x = 0

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જવાબ : Given :

y2 − 4y + 4x = 0

⇒(y−2)2−4+4x=0

⇒(y−2)2=−4(x−1)
Let Y=y−2, X=x−1
Then, we have:
Y2=−4X

Comparing the given equation with Y2=−4aX:

4a=4

⇒a=1

∴ Vertex = (X = 0, Y = 0) = (x=1, y=2)x=1, y=2


Find the vertex of the following parabolas  :  y2 − 4y − 3x + 1 = 0

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જવાબ : Given: 
y2 − 4y − 3x + 1 = 0

⇒(y−2)2−4−3x+1=0

⇒(y−2)2=3(x+1)

⇒(y−2)2=3(x−(−1))

Let Y=y−2, X=x+1
Then, we have:
Y2=3X

Comparing the given equation with Y2=4aX:

4a=3

⇒a=34

∴ Vertex = (X = 0, Y = 0) = (x=−1, y=2)


Find the vertex of the following parabolas  :  4x2 + y = 0

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જવાબ : Given:
  4x2 + y = 0
⇒-y/4=x2

On comparing the given equation with x2=−4ay:

4a=1/4

⇒a=1/16

∴ Vertex = (0, 0)


Find the vertex of the following parabolas  :  y2 = 8x

 

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જવાબ : Given:
  y2 = 8x

On comparing the given equation with y2=4ax:
4a=8⇒a=2

∴ Vertex = (0, 0)


Find the radius of each of the following circles :  x2 + y2 − x + 2y − 3 = 0.

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જવાબ : Let (p, q) be the centre of a circle with radius a.
Thus, its equation will be (x−p)2 + (y−q)2 = a2

Given:
x2 + y2− x + 2y − 3=0

The given equation can be rewritten as follows:
(x− ½ )2+(y+1)2− ¼ −1−3=0
⇒(x− ½ )2 + (y+1)2 = 17/4

Thus, the centre is ( ½ ,−1) and and the radius is √17/2.


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