Practice Question on conic section for IIT-Jee, NDA and Airforce

Conic Section

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Question no. 1

The points on the parabola y² = 36x whose ordinate is three times the abscissa are

looks_one (0,0), (4,12)

looks_two (1,3) ,(4,12)

looks_3 (4,12)

looks_4 none of these

Option looks_one (0,0), (4,12)

Solution :
ordinate → y
abscissa → x
y = 3x
y² = 36x ⇒ 9x² = 36x
9x(x-4) = 0 ⇒ x =0 , 4
y = 0 , 12
points are (0,0) and (4,12)

Question no. 2

The focal distance of a point on the parabola y² = 16x whose ordinate is twice the abscissa , is

looks_one 6

looks_two 8

looks_3 10

looks_4 12

option looks_two 8

Solution :

y²= 4ax
y = 2x
y² = 16x
4x² = 16x ⇒ x= 4 , 0
y = 8 ,0
focal point of given parabola = (4,0)
distance between between (4,8) and (4,0) is 8

Question no. 3

The focus of the parabola x²=-16y is

looks_one (4,0)

looks_two (0,4)

looks_3 (-4 ,0)

looks_4 (0,-4)

option looks_4 (0,-4)

Solution :

x²=-16y
x²=-4ay ⇒ focus =(0 , -a )
x²=-16y ⇒ a = 4
focus = (0 ,-4)

Question no. 4

If (2,0) is the vextex and y axis the directrix of a parabola ,then its focus is

looks_one (2,0)

looks_two (-2,0)

looks_3 (4,0)

looks_4 (-4,0)

option looks_3 (4,0)

Solution :

y axis is the directrix of given parabola , it means given parabola is symmetric along x axis
directrix distance from the vertex is equal to the distance of focus from the vertex
directrix is 2 unit from vertex , so focus will be 2 unit from vertex
vertex point = (2 ,0) ⇒ focus point = (4 ,0)

Question no. 5

The ends of latus rectum of parabola x² + 8y =0

looks_one (-4 ,-2) and (4,2)

looks_two (4 ,-2) and (-4,2)

looks_3 (-4 ,-2) and (4,-2)

looks_4(-4 ,2) and (4,2)

option looks_3 (-4 ,-2) and (4,-2)

Solution :

x² + 8y =0 ⇒ x² = -8y
a = 2
focus point = (0,-2)
latus rectum ends point = (x₁, -2) , (x₂ , -2)
put the points in the equations
x² = -8y ⇒ x² = 16 ⇒ x = 4 ,-4
points of latus point = (-4 ,-2) and (4,-2)

Question no. 6

Foci and directrices of the parabola x² = -8ay are

looks_one (0,-2a ) and y=2a

looks_two (0,-2a ) and y=-2a

looks_3 (2a,0 ) and x=-2a

looks_4 (-2a ,0 ) and x=2a

option looks_one (0,-2a ) and y=2a

Solution :

x² = -8ay (x² = -4Ay) ⇒ A = 2a
foci : (0 , -2a)
directrix : y = 2a

Question no. 7

The ends of latus rectum of parabola x²= 4ay.

looks_one (a ,2a), (2a ,-a)

looks_two (-a ,2a), (2a ,a)

looks_3 (a ,-2a), (2a ,a)

looks_4 (-2a ,a), (2a ,a)

option looks_4 (-2a ,a), (2a ,a)

Solution :

latus rectum = (-2a, a), (2a ,a)

Question no. 8

The equation of the parabola with focus(3,0) and the directrix x+3=0

looks_one y²= 3x

looks_two y²= 2x

looks_3 y²= 12x

looks_4 y²= 6x

option looks_3 y²= 12x

Solution :

focus = (3,0)
directrix x+3 = 0 ⇒ a = 3
parabola is symmetrical to x axis
y² = 4ax
put the value of a = 3
y² = 4*3*x = 12x

Question no. 9

The parabola y²= x is symmetric about

looks_one y-axis

looks_two x-axis

looks_3 both x-axis and y-axis

looks_4 the line y=x

option looks_two x-axis

Solution:

The parabola y²= x is symmetric about x-axis

Question no. 10

Vertex of the parabola y² + 2y +x= 0 lies in the quadrant

looks_one first

looks_two second

looks_3 third

looks_4 fourth

option looks_4 fourth

Solution :
y² + 2y +x= 0 ⇒ (y+1)² = -x + 1 = -(x-1)

Question no. 11

If the latus rectum of an ellipse is equal to half its minor axis, then what is its eccentricity?

looks_one 1/2

looks_two √3

looks_3 √3/2

looks_4 1/√2

option looks_3 √3/2

Solution :

Question no. 12

What are the points of intersection of the curve 4x² - 9y² = 1 with its conjugate axis?

looks_one (1/2 ,0) and (-1/2 ,0)

looks_two (0,2) and (0,-2)

looks_3 (0,3) and (0,-3)

looks_4 No such point exist

option looks_4 No such point exist

Solution :
conjugate axis never cut the hyperbola , therefore no such point exit

Question no. 13

What is the sum of the local distances of a point of an ellipse x²/a² + y²/b² =1 ?

looks_one a

looks_two b

looks_3 2a

looks_4 2b

option looks_3 2a

Solution :

distance of f'A and fA = sum of local points of a ellipse
f'A = a + ae
fA = a-ae
f'A + fA = 2a

Question no. 14

What is the eccentricity of the conic 4x² +9y² = 144.

looks_one √5/3

looks_two √5/4

looks_3 3/√5

looks_4 2/3

option looks_one √5/3

Solution :

Question no. 15

The sum of the focal distances of a point on the ellipse x²/4 + y²/9 = 1 is

looks_one 4

looks_two 6

looks_3 8

looks_4 10

option looks_two 6

Solution :

sum of the focal distances = 2b = 2*3 = 6

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