## Kerala Plus One Maths Chapter Wise Previous Questions and Answers Chapter 11 Conic Sections

Question 1.

Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16. [March-2018]

Answer:

Let the equation of the circle be (x – h)^{2} + (y -k)^{2} = r^{2 }Since the circle passes through (4, 1) and (6,5), we have

(4-h)^{2} + (1-k)^{2} = r^{2}…………. (1)

and (6 -h)^{2} + ( 5 – k)^{2} = r^{2}…………. (2)

Also since the centre lies on the line 4x + y = 16, we have

4h + k= 16…………… (3)

Solving the equations (1), (2) and (3), we get

h = 3 and k = 4 (4-3)^{2} + (1 -4)^{2} = r^{2 }t^{2}= 10

Hence, the equation of the required circle is

(x-3)^{2}+(y-4)^{2} = 10

x^{2}-6x + 9 + y^{2}-8y+16-10 = 0

x^{2} + y^{2}-6x-8y+ 15 = 0

Question 2.

The figure shows an ellipse and line L.

a. Find the eccentricity and focus of the ellipse.

b. Find the equation of the line L.

c. Find the equation of the line parallel to line L and passing through any one of the foci. [March-2018]

Answer:

c. Required equation is 3x + 5y+k=0…………… (1)

Since it passes through the focus (4,0), we have 3(4) + 5(0) + k = 0

⇒ k=-12

In (1),3x + 5y-12 = 0

Question 3.

a. Find the equation of the parabola with focus (6, 0) and equation of the directrix is x = – 6.

b. Find the coordinates of the foci, the vertices, the length of transverse and conjugate axis and eccentricity of the hyperbola

[March-2017]

Answer:

Question 4.

Find the foci, vertices, length of the major axis and eccentricity of the ellipse:

Answer:

Question 5.

Directrix of the parabola x^{2} = – 4ay is…..

i. x + a = 0

ii. x – a = 0

iii. y – a = 0

iv. y + a = 0

b. Find the equation of the ellipse whose length of the major axis is

20 and foci are (0, ± 5). [March-2015]

Answer:

Question 6.

Consider the ellipse

Find the coordinates of the foci, the length of the major axis, the length of the minor axis, latus rectum and eccentricity. [March-2014]

Answer:

Question 7.

Find the coordinates of the foci, the length of the major axis, minor axis, latus rectum and eccentricity of the ellipse [March-2013]

Answer:

Question 8.

i. Find the centre and radius of the circle x^{2}+ y^{2} – 4x – 8y – 45 = 0.

ii. Find the foci the eccentricity and the length of the latus rectum of the ellipse 4x^{2} +9y^{2 }= 36.

iii. Find the equation of a parabola with vertex at (0,0) and ficus at (0,2). [February-2013]

Answer:

i. x^{2}+y^{2} – 4x – 8y – 45 = o

⇒ x^{2} – 4x + y^{2} – 8y = 45

⇒ (x^{2}-4x + 4) + (y^{2}-8y+16) = 45 +16 + 4

⇒ (x-2)^{2} + (y-4)^{2} = 65

.’. Radius = units

ii. 4x^{2} + 9y^{2} = 36 divided by 36

iii.Since the vertex is at (0,0) and the focus is at (0,2) which lies on y-axis, the y-axis is the axis of the parabola. Therefore, equation of the parabola is of the form x^{2} = 4 a y. Thus, we have x^{2} = 4 (2) y,

i.e.,x^{2} =8y

Question 9.

A hyperbola whose transverse axis is X- axis, center (0,0) and the foci (± , 0) passes through the point (3,2).

a. Find the equation of the hyperbola.

b. Find its eccentricity. [March-2012]

Answer:

Question 10.

i. Find the equation of the circle with center (2,2) and passing through the point (4,5).

ii. Find the eccentricity and the length of latus rectum of the ellipse 4x^{2} + 9y^{2} =36 [March-2011]

Answer:

ii. 4x^{2} + 9y^{2} = 36 divided by 36

Question 11.

An ellipse whose major axis as X-axis and the centre (0, 0) passes through (4, 3) and (-1,4).

i. Find the equation of the ellipse.

ii. Find its eccentricity. [March-2010]

Answer:

Question 12.

Consider the conic 9y^{2} – 4x^{2} = 36. Find:

a. The foci

b. Eccentricity.

c. Length of latus rectum.[September-2010]

Answer:

Question 13.

Consider the circle x^{2} + y^{2} + 8x +10y-8= 0

i. Find its centre C and radius

ii. Find the equation of the circle with centre at C and passing through the point (1,2). [August-2009]

Answer:

i. The given equation is (x^{2} + 8x) + (y^{2} +1oy) = 8

Now, completing the squares within the parenthesis, we get

(x^{2} +8x+ 16) + (y^{2}+ 10y + 25) = 8+ 16 + 25

i.e. (x + 4)^{2} + (y + 5)^{2} = 49

i.e. (x-(-4)}^{2} + {y – (-5)}^{2} = 7^{2 }Therefore, the given circle has centre at (- 4, -5) and radius 7

ii. Radius = distance between (- 4, -5) and (1,2)=

Equation of the circle is (x + 4)^{2} +(y +5)^{2 }= 74

x^{2} + y^{2} + 8x + 10y – 33 = 0

Question 14.

i. Find the equation of the parabola with vertex at (0,0) and focus at (0,2).

ii. Find the co-ordinates of the foci and the latus rectum of the ellipse

[August-2009]

Answer:

Since the vertex is at (0,0) and the focus is at (0,2) which lies on y-axis, the y-axis is the axis of the parabola. Therefore, equation of the parabola is of the form x^{2} = 4 a y.

Thus, we have x^{2} = 4 (2) y,i.e.,x^{2} =8y

Question 15.

a. Find the centre and radius of the circle 2x^{2}+ 2y^{2} – x = 0

b. Find the equation of the parabola with focus (6,0) and directrix x =-6. [March-2009]

Answer:

a.

b. a = 6,

Equation is y^{2} = 4ax

i.e,y^{2} = 24x

Question 16.

Find the coordinates of the foci, vertices and the length of latus rectum of the ellipse

[March – 2009]

Answer:

a = 5, b = 10; Here a < b

a^{2} = b^{2}(1 – e^{2}),

25 = 10(1 -e^{2}),

Question 17.

Find the equation of the circle concentric with the circle x^{2} + y^{2} – 4x – 6y – 9 = 0 and passing

through (- 4, -5). [September-2008]

Answer:

x^{2} + y^{2} – 4x – 6y – 9 = 0

x^{2} – 4x + y^{2}– 6y = 9

(x^{2} – 4x + 4) + (y^{2}– 6y + 9) = 9 + 4 + 9 (x-2)^{2} + (y-3)^{2} = 22

Centre: (2,3); Radius =

Radius of required circle

∴ Equation of required circle is

(x-2)^{2} + (y-3)^{2} = 100

Question 18.

a. Find the parabola x^{2} = -4ay, the focus is at

i. (0, 0) ii. (a, 0) iii. (-a, 0) iv. (0, a)

b. y^{2} = -12x, represents a parabola whose directrix is

c. Find the equation of the ellipse with centre at the origin, length of major axis is 12 and focus at (4,0).

d. 9 x^{2} – 16y^{2} = 144 is a hyperbola with eccentricity [September-2008]

Answer:

a.(0, -a) x^{2}=-4ay

Equation of axis is y-axis

Company with equation x^{2} = 4ay; a= -a

focus of the parabola (0, -a)

Question 19.

If A (-2,3), B (3, -5).

Find the equation of the circle with AB as diameter. [June-2008]

Answer:

Question 20.

i. An ellipse has its centre at origin, whose major axis is 5 and minor axis is 4

a. Write the equation.

b. What is its eccentricity?

ii. For the parabola y^{2} = 8x write its focus and equation of its directrix.

iii. Find the equation of the hyperbola whose vertices are (± 5,0) and foci (± 8,0).

Answer:

Question 21.

i. Represent the ellipse x^{2} + 16y^{2} = 16 in standard form

ii. What is its eccentricity? [September-2007]

Answer:

Question 22.

Consider the figure given below.

i. Find the centre of the circle.

ii. Find the equation of the circle. [June-2007]

Answer:

i. Solving 3x – y = 2 and x + 2y = 3

We getx=1,y=1

∴ Centre is (1,1)

ii. Radius =

Equation is (x -1 )^{2} + (y -1 )^{2} = 2

Question 23.

Consider the ellipse 16x^{2} + 9y^{2} = 144

i. On which coordinate axis does the foci of this ellipse lie.

ii. Determine the eccentricity of the ellipse. [February-2007]

Answer:

Question 24.

i. Consider the equation x^{2} = -4ay. Its graph is

ii. Find the coordinates of the focus and length of the latus rectum of the parabola y^{2}=-6x [June-2006]

Answer:

i. Parabola open downwards

ii. This is of the form y^{2} = – 4ax.

4a = 6 or a = 3/2

∴ focus is (-a, 0) i.e, (-3/2,0)

Length of latus rectum = 4a = 4 × = 6

Question 25.

In the figure, S is the focus, O is the vertex and the line DD’ is the directix of the parabola x^{2}=-4ay.

i. Write down the length of the latus rectum.

ii. Find the coordinates of the focus S and the equation of the directrix DD’. [March-2006]

Answer:

i. Length of latus rectum = 4a

ii. Focus is (0, -a)

iii y = a