Conic section for IIT-JEE . NDA and Airforce 16-30( SET 2)

conic section

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Question no. 16

P is any point on the ellipse 9x² + 36y² = 324, whose foci are S and S' . Then SP + S'P equals

looks_one 3

looks_two 12

looks_3 36

looks_4 324

Option looks_two 12

solution :
9x² + 36y² = 324
x²/36 + y²/16 = 1
SP + S'P = 2a = 2*6 = 12

Question no. 17

The equation of the ellipse whose vertices are (∓5,0) and foci at (∓4, 0) is

looks_one x²/25 + y²/9 =1

looks_two x²/9 + y²/25 =1

looks_3 x²/16 + y²/25 =1

looks_4 x²/25 + y²/16 =1

option looks_one x²/25 + y²/9 =1

solutw2ion :
a= 5
ae = 4
e = 4/5
e² = 1-b²/a²
16/25 = 1-b²/25
b = 3
the equation of ellipse , x²/25 + y²/9 =1

Question no. 18

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

looks_one latus rectum

looks_two semi-transverse axis

looks_3 transverse axis

looks_4 semi-latus rectum

option looks_3 transverse axis

Solution :
The difference of focal distances of any point on hyperbola is equal to transverse axis

Question no. 19

The equation x² -16xy -11y² -12x +6y +21=0 represents

looks_one parabola

looks_two ellipse

looks_3 hyperbola

looks_4 two straight lines

option looks_3 hyperbola

Solution :
h= -8 ⇒ h² = 64
a= 1
b = -11
h² > ab and abc + 2fgh - af² - bg² - ch² ≠ 0
therefore it is hyperbola

Question no. 20

The equation of the hyberbola whose directrix is 2x+y =1 , focus (1,1) and eccentricity = √3 , is

looks_one 7x² +12xy -2y² -2x + 4y -7 =0

looks_two 11x² +12xy +2y² -10x - 4y +1 =0

looks_3 11x² +12xy +2y² -14x - 14y +1 =0

looks_4 none of these

option looks_one 7x² +12xy -2y² -2x + 4y -7 =0

solution :
SP/SQ = e

5(x² +1 -2x + y² +1 - 2y) = 3 (4x² + y² +1 - 4x - 2y + 4xy )
7x² +12xy -2y² -2x + 4y -7 =0

Question no. 21

x² -4y² -2x +16y + 40 = 0 represents

looks_one a pair of straight lines

looks_two ellipse

looks_3 hyperbola

looks_4 parabola

option looks_3 hyperbola

Solution :
abc + 2fgh - af² - bg² - ch² ≠ 0
h² > ab
therefore , it is hyperbola

Question no. 22

The distance between the directrices of the hyperbola x = 8secθ and y= 8tanθ

looks_one 16√2

looks_two √2

looks_3 8√2

looks_4 4√2

option looks_3 8√2

Solution :
x = asecθ and y = btanθ
a = 8 , b = 8
e²= 1 + a²/b²
e = √2
x = a/e = 8/√2
the distance between directrices = 2 *8/√2 = 8/√2

Question no. 23

The eccentricity of the hyperbola 5x² - 4y² +20x +8y - 4=0 is

looks_one √2

looks_two 3/2

looks_3 2

looks_4 3

option looks_two 3/2

Solution :
5x² - 4y² +20x +8y - 4=0 , this equation can be written as
(x+2)²/4 - (y-1)²/5 = 1

Question no. 24

Consider the parabola S₁ = y² - 4ax =0 , and S₂ = y² - 4bx=0 . S₂ will contain S₁ , If

looks_one a > b > 0

looks_two b > a > 0

looks_3 a >0 , b<0 but |b|>a

looks_4 a <0 , b> 0 but b > |a|

option looks_two b > a > 0

Solution :

S₁ = y² - 4ax =0
S₂ = y² - 4bx=0 . S₂ will contain S₁
a > 0 , b> 0
latus rectum of S2 will always greater than S1 , therefore 4b > 4a
b > a

Question no. 25

What the sum of focal radii of any point on an ellipse equal to?

looks_one length of latus rectum

looks_two length of major axis

looks_3 length of minor axis

looks_4 length of semi latus rectum

option looks_two length of major axis

Sum of focal radii of any point on an ellipse equal to is length of major axis

Question no. 26

What is the locus of points , the difference of whose distances from two points being constant?

looks_one hyperbola

looks_two ellipse

looks_3 parabola

looks_4 pair of straight lines

option looks_one hyperbola

the locus of points when the difference of whose distances from two points being constant is hyperbola

Question no. 27

What are the equations of the directrices of the ellipse 25x² + 16y² =400?

looks_one 3x ∓ 25= 0

looks_two x ∓ 15=0

looks_3 3y ∓ 25= 0

looks_4 y ∓ 25= 0

option looks_3 3y ∓ 25= 0

Soltion :
25x² + 16y² =400
x²/16 + y²/25 =1
a<b : so equation of directrix is y = ∓b/e

e = 3/5
y = ∓b/e
y = ∓25/3
3y ∓ 25= 0

Question no. 28

Let E be the ellipse x²/9 + y²/4 = 1 and C be the circle x² + y² =9 . Let P (1,2) and Q (2,1) . Which one of the following is correct?

looks_one Q lies inside C but outside E

looks_two Q lies outside both C and E

looks_3 P lies inside both C and E

looks_4 P lies inside C but outside E

option looks_4 P lies inside C but outside E

solution :
Point P (1,2) :
circle : x² + y² =9
1² + 2² = 5 < 9 , so point is inside the circle
ellipse : x²/9 + y²/4 = 1
1²/9 + 2²/4 = 10/9 > 1 , so point is outside the ellipse
Point P (2,1) :
circle : x² + y² =9
2² + 1² = 5 < 9 , so point is inside the circle
ellipse : x²/9 + y²/4 = 1
2²/9 + 1²/4 = 25/36 < 1 , so point is inside the ellipse

Question no. 29

A point moves such that difference of its distances from two given points (c,0) and (-c,0) is constant. What is the locus of the point P?

looks_one circle

looks_two ellipse

looks_3 hyperbola

looks_4 parabola

option looks_3 hyperbola

Solution :
the locus of point is hyperbola

Question no. 30

The curve y²= -4ax (a>0) lies in

looks_one first and fourth quadrants

looks_two first and second quadrants

looks_3 second and third quadrants

looks_4 third and fourth quadrants

option looks_3 second and third quadrants

solution :

second and third quadrants

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