1 CONIC SECTION ASSIGNMENT -6 (DATE:- 20.04.2020) SINGLE CORRECT 1 2 3 4 Two mutually perpendicular tangents of the parabola y 2 = 4ax (a > 0) meet the axis in P 1 and P 2 . If S is the focus of the parabola then 1 2 1 1 l( ) l( ) SP SP is equal to 1) 4 a 2) 2 a 3) 1 a 4) 1 4a 5 The straight line joining any point P on the parabola y 2 = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is 1) x 2 + 2y 2 – ax = 0 2) 2x 2 + y 2 – 2ax = 0 3) 2x 2 + 2y 2 – ay = 0 4) 2x 2 + y 2 – 2ay = 0
Transcript
1
CONIC SECTION
ASSIGNMENT -6 (DATE:- 20.04.2020)
SINGLE CORRECT
1
2
3
4 Two mutually perpendicular tangents of the parabola y2 = 4ax (a > 0) meet the axis in
P1 and P2. If S is the focus of the parabola then 1 2
1 1
l( ) l( )SP SP is equal to
1)4
a 2)
2
a 3)
1
a 4)
1
4a
5 The straight line joining any point P on the parabola y2 = 4ax to the vertex and
perpendicular from the focus to the tangent at P, intersect at R, then the equation of
the locus of R is
1) x2 + 2y2 – ax = 0 2) 2x2 + y2 – 2ax = 0
3) 2x2 + 2y2 – ay = 0 4) 2x2 + y2 – 2ay = 0
2
MULTY CORRECT TYPE
6. If a circle touches the parabola 2 4y x= at A(9,6) and is tangent to x-axis then which is/are correct
A) The minimum possible radius of circle is 60 18 10-
B) The maximum possible radius of circle is 60 18 10+
C) The y-intercept of common normal through A is 3 times x-intercept of common normal through A
D) The area of triangle formed by tangent, normal at A to parabola and x-axis is 60 square units
7. Let ‘O’ be origin and the curve y x 9 meets the x-axis and y-axis respectively at A and B. Which of
the following is/ are correct
A) The abscissa of the point where the tangent at B intersect the curve again is 36 18 2
B) Normal at A to the curve is x + 9 = 0
C) Tangent at B meets the tangent at A at , such that =15
2
D) The area of quadrilateral formed by the tangents at A and B and the co-ordinate axes is 33
4 square units
8. Let lines 1 1L : y 6 m x 4 and 2 2L : y 6 m x 4 touching parabola 2y 4ax a 0 at points
A and B respectively where A lies in I quadrant and 1 2
1 1 3,
m m 2 then
A) Length of latus rectum of parabola is 12
B) 1L and 2L are perpendicular
C) 1 2m m
D) Slope of normal at point B on parabola is 2
9. If a tangent of slope 1
3of the ellipse
2 2
2 21
x y
a b+ = ( )a b> is normal to the circle
2 2 2 2 1 0x y x y+ + + + = then which is/are correct
A) maximum value of ab is 2
3 B)
2,2
5a
æ ö÷ç ÷Î ç ÷ç ÷çè ø
C) 2
,23
aæ ö
÷çÎ ÷ç ÷çè ø D) Maximum value of ab is 1
10. If a tangent of slope 1
3 of the ellipse
2 2
2 21
x y
a b (a > b) is normal to the circle 2 2 2 2 1 0x y x y ,
then
3
A) Maximum value of ab is 2
3 B) 2
,25
a
C) 2,2
3a
D) maximum value of ab is 1
COMPREHENSION TYPE QUESTIONS
Paragraph for Question Nos. 11 to 12
y f x is parabola of the form 2 1f x x bx , b is a constant. The tangent line is drawn at the
point where f x cuts y – axis, also touches2 2x y r . It is also given that at least one tangent can be
drawn from point P to y f x , where D is a point at which y x is non – differentiable .R
11. For maximum value of b, the area of circle is
A) 10
B)
5
C) D) 5
12. The value of L is,.
A) 21 1,1 , 1,0y x x y
B) 21 2,2 , 0,1y x x y
C) 21 2,2 , 3,1y x x y
D) 21 1,1 , 0,1y x x y
4
INTEGER TYPE QUESITNS 15. An isosceles triangle has its vertex at origin and its base, parallel to the x-axis with the vertices above
the x-axis on the curve 227y x . Then maximum area of the triangle is K square units, where9
k
( [.]
G.I.F )
16. Radius of the circle which touches the tangents drawn from ( -2, 0) to the parabola 2y 4x & the parabola
2y 4x is , then 2
2 1 ______
17. The shortest distance between the parabola 22 2 1y x= - and
22 2 1x y= - is 1
d then
2d ___
18. Area of the figure (polygon) formed by the points on the ellipse
2 2
150 20
x y+ = , such that the pair of
tangents drawn from each of such points to the ellipse
2 2
116 9
x y+ = are perpendicular is K then sum of
digits of K equals
19. Let L be locus of midpoint of focal chords of the ellipse2 2x y
19 25 . The integer nearest to the area
enclosed by curve L is ................
20. The three ellipses
2 2
2 2
i i
x y1
a b , i = 1, 2, 3, have a common tangent, then the value of
2 2
1 1
2 2
2 2
2 2
3 3
a b 1
a b 1
a b 1
5
MATRIX MATCHING TYPE QUESTIONS
21. A line L: 3y mx meets the y-axis at E(0,3) and the arc of the parabola
2 16 , 0 6y x y , at the point 0 0,F x y . The tangent to the parabola at 0 0,F x y
intersects the y-axis at 10,G y . The slope m of the line L is chosen such that the area
of triangle EFG has a local maximum.
Column-I Column-II
A) m= P) ½
B) Maximum area of EFG is Q) 4
C) 0y = R) 2
D) 1y = S) 1
22. If AB is a chord subtending right angle at the vertex of 2 4y x , P is the centroid of
OAB and S be the focus of parabola (where O is origin)
Column-I Column-II
A) The min area of PAB is P) 4
3
B) The locus of centroid of SAB is a
parabola whose length of
latusrectum is 4
then is
Q) 10
3
C) If the focus of the locus of centroid of
SAB is , then is
R) 16
3
D) The distance of P from the directrix of
2 4y x when area of PAB is
S) 11
3
6
minimum is
KEY AND SOLUTIONS
1 2 3 4 5 6 7 8 9
A B D 3 2 ABCD AC BCD AB
10 11 12 13 14 15 16 17 18
AB B D BCD AD 6 6 8 5
19 20 21 22
8 0 A-S
B-P
C-Q D-R
A-R
B-R
C-Q D-S
SINGLE CORRECT
1
2
7
3
4
Key: 3
SP1 = a(1 + 2
1t ) ;SP2 = a(1 + 2
2t )
t1t2 = – 1
1
1
SP =
2
1
(1 )a t ;
2
1
SP =
2
2(1 )
t
a t
1
1
SP +
2
1
SP=
1
a
5 Key: 2
T :ty = x + at2 ....(1)
8
line perpendicular to (1) through (a,0)
tx + y = ta .... (2)
equation of OP : y –2
t x = 0....(3)
from (2) & (3) eleminating t we get locus ]
MULTY CORRECT TYPE 6. KEY: ABCD
SOL: Minimum 3
610
r ræ ö
÷ç- =÷ç ÷÷çè ø
60 18 10rÞ = -
Maximum 3
610
r ræ ö
÷ç+ =÷ç ÷÷çè ø
60 18 10rÞ = +
Tgt: 3 9 0x y- + =
Normal 3 33 0x y+ - =
111 33
x yÞ + =
Area 26 1 10
3 18 602 3 3
= + = ´ = sq. units
7. Key: AC
SOL:
Tgt at B
x-6y+18=0
Solving with 2y x 9
2y 6y 18 9
2y 6y 9 0 y 3 3 2 x 36 18 2
Solving x - 6y + 18 = 0 and x + 9 = 0 3 15
92 2
,
Area
391 1 27 81
27292 2 2 4
3
8. KEY: BCD
9. If a tangent of slope 1
3of the ellipse
2 2
2 21
x y
a b+ = ( )a b> is normal to the circle
2 2 2 2 1 0x y x y+ + + + = then which is/are correct
9
A) maximum value of ab is 2
3 B)
2,2
5a
æ ö÷ç ÷Î ç ÷ç ÷çè ø
C) 2
,23
aæ ö
÷çÎ ÷ç ÷çè ø D) Maximum value of ab is 1
KEY: AB
SOL: Conceptual
10. KEY: AB
SOL: 2 2 2y mx a m b 2
21 ay x b
3 9
2 23y x a 9b passes through (-1, -1)
2 23 1 a 9b
2 22 a 9b 2 29 4a b 2 2
2 2a 9b9a b
2
COMPREHENSION TYPE QUESTIONS 11. KEY: B
12. KEY: D
13. KEY: BCD
14. KEY: AD
SOL: 57 & 58
10
INTEGER TYPE QUESITNS 15. KEY: 6
SOL: K = 54
16. KEY: 6
SOL:
12 2 2
2r 3 1S 2 6
2
r 1 3
17. KEY: 8
SOL:
P
Qy x
3 1
,4 2
Pæ ö
÷ç= ÷ç ÷çè ø
11
1 3
,2 4
Qæ ö
÷ç= ÷ç ÷çè ø
1
2 2PQ =
18. KEY: 5
SOL: circle 2 2 25x y+ =
There are 4 points of intersect with given ellipse
