Circle and Conic Section (Solved MCQs and Notes)

Notes 49 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

92
Unit – 11 – Circle and Conic Section
MCQ
(1) The number of integral values of m for which x
2
+ y
2
(1 – m)x + my + 5 = 0 is the equation
of a circle whose radius cannot exceed 5, is
(a) 20 (b) 18 (c) 8 (d) 24
(2) The circle x
2
+ y
2
– 6x – 10y + ? = 0 does not touch or intersect the coordinate axes and
point (1, 4) is inside the circle, then the range of the values of ? is
(a) (0, 25) (b) (5, 29] (c) (25, 29) (d) (9, 25)
(3) Equation of smallest circle touching these four circle (x ? 1)
2
+ (y ? 1)
2
= 1 is
(a) x
2
+ y
2
= 3 –
2
(b) x
2
+ y
2
= 5 – 2
2
(c) x
2
+ y
2
= 6 – 2
2
(d) x
2
+ y
2
= 3 – 2
2
(4) If two circle (x – 1)
2
+ (y – 3)
2
= a
2
and x
2
+ y
2
– 8x + 2y + 8 = 0 intersect in two distinct
points, then
(a) 2 < a < 8 (b) a > 2 (c) a < 2 (d) a = 2
(5) If the tangents are drawn to the circle x
2
+ y
2
= 12 at the point where it meets the circle
x
2
+ y
2
– 5x + 3y – 2 = 0, then the point of intersection of these tangent is
(a) (6, – 6) (b)
6
18
5
,
F
H
I
K
(c)
6
18
5
,
?
F
H
I
K
(d)
?
?
F
H
I
K
6
18
5
,
(6) Two tangents to the circle x
2
+ y
2
= 4 at the points A and B meet at P(–4, 0). The area of the
quadrilateral PAOB, where O is the origin is
(a)
4 3
(b) 4 (c)
6 2
(d)
2 3
(7) The radius of the circle passing through the points (5, 2), (5, –2) and (1, 2) is
(a)
2 5
(b) 3
2
(c) 5
2
(d) 2
2
(8) The line xsin? – ycos? = k touches the circle x
2
+ y
2
= k
2
then
(a)
?
? ?
? ?
L
N
M
O
Q
P
2 2
,
(b) ? ? [0, ? ] (c) ? ? [–? , ? ] (d) ? is any angle
(9) One of the diameters of the circle circumscribing the rectangle ABCD is x – 4y + 7 = 0. If
A and B are points (–3, 4) and (5, 4) respectively, then the area of the rectangle is
(a) 32 sq. units (b) 16 sq. units (c) 64 sq. units (d) 8 sq. units
(10) Let C be the centre of the circle x
2
+ y
2
– 2x – 4y – 20 = 0. If the tangents at the point A(1, 7)
Page 1
93
and B(4, –2) on the circle meet at piont D. Then area of the quadrilateral ABCD is
(a) 150 sq. units (b) 100 sq. units (c) 75 sq. units (d) 50 sq. units
(11) The circle x
2
+ y
2
– 4x – 4y + 4 = 0 is inscribed in a triangle which has two of its sides along
the co–ordinates axes. The locus of the circumcentre of the triangle is x + y – xy + k
x y
2 2
0? ?
then k =
(a) 0 (b) ? 1 (c) 2 (d) ? 3
(12) A square is inscribed in the circle x
2
+ y
2
– 2x + 4y + 3 = 0. Its sides are parallel to the co–
ordinate axes. Then one vertex of the square is
(a) 1 2 2? ?,
c
h
(b) 1 2 2? ?,
c
h
(c) (1, -2+
2
) (d) 1 2 2 2? ? ?,
c
h
(13) If the equation
m x y( ) ( )?
?
?
?
1
3
2
4
1
2 2
represents a circle then m =
(a) 0 (b)
3
4
(c)
?
3
4
(d) 1
(14) The circle whose equation is x
2
+ y
2
– 2? x – ? y + ?
2
= 0
(a) passes through origin (b) touches only X–axis
(c) touches only Y–axis (d) touches both the axes
(15) The line (x + g) cos? + (y + f ) sin? = k touches the circle x
2
+ y
2
+ 2gx + 2fy + c = 0 only
its
(a) g
2
+ f
2
= c + k
2
(b) g
2
+ f
2
= c
2
+ k
2
(c) g
2
+ f
2
= c – k
2
(d) g
2
+ f
2
= c
2
– k
2
(16) The centre of the circle passing throug (0, 0) and (1, 0) and touching the circle x
2
+ y
2
= 9
is
(a)
3
2
1
2
,
F
H
I
K
(b)
1
2
3
2
,
F
H
I
K
(c)
1
2
1
2
,
F
H
I
K
(d)
1
2
2, ?
F
H
I
K
(17) The number of common tangents to the circles x
2
+ y
2
= 4 and x
2
+ y
2
– 6x – 8y – 24 = 0
is
(a) 0 (b) 1 (c) 2 (d) None of these
(18) The equation of the set of complex number z = x + iy, So that | z – z
1
| = 5, where z
1
= 1 + 2i
(a) x
2
+ y
2
– 2x – 4y – 20 = 0 (b) x
2
+ y
2
+ 2x – 4y – 20 = 0
(c) x
2
+ y
2
– 2x + 4y – 20 = 0 (d) x
2
+ y
2
+ 2x + 4y + 20 = 0
Page 2
94
(19) A circle is given by x
2
+ (y – 1)
2
= 1, another circle C touches it externally and also the x–
axis, then the locus of its centre is
(a) {(x, y) : x
2
= 4y} ? {(x, y) : y ? 0} (b) {(x, y) : x
2
+ (y – 1)
2
= 4} ? {(x, y) : y ? 0}
(c) {(x, y) : x
2
= y} ? {(0, y) : y ? 0} (d) {(x, y) : x
2
= 4y} ? {(0, y) : y ? 0}
(20) Tangent to the circle x
2
+ y
2
= 5 at the point (1, –2) also touches the circle x
2
+ y
2
– 8x +
6y + 20 = 0 then point of contact is
(a) (3, 1) (b) (3, –1) (c) (–3, –1) (d) (–3, 1)
(21) Four distinct points (1, 0), (0, 1), (0, 0) and (2a, 3a) lie on a circle for
(a) only one value of a ? (0, 1) (b) a > 2
(c) a < 0 (d) a ? (1, 2)
(22) The length of the chord joining the points (2cos? , 2sin? ) and (2cos(? + 60
o
), 2sin(? + 60
o
))
of the circle x
2
+ y
2
= 4 is
(a) 2 (b) 4 (c) 8 (d) 16
(23) A square is formed by the two points of straight lines x
2
– 8x + 12 = 0 and y
2
– 14y + 45
= 0. A circle is inscribed in it. The centre of the circle is
(a) (6, 5) (B) (5, 6) (c) (7, 4) (d) (4, 7)
(24) If one of the diameters of the circle x
2
+ y
2
– 2x – 6y + 6 = 0 is a chord to the circle with
centre (2, 1), then the radius of the circle is
(a) 3 (b)
3
(c) 2 (d)
2
(25) The lines 2x – 3y – 5 = 0 and 3x – 4y – 7 = 0 are diameters of a circle of area 154 square
units then the equation of the circle is
(a) x
2
+ y
2
+ 2x – 2y – 62 = 0 (b) x
2
+ y
2
+ 2x – 2y – 47 = 0
(c) x
2
+ y
2
– 2x + 2y – 47 = 0 (d) x
2
+ y
2
– 2x + 2y – 62 = 0
(26) The equation of the common tangent to the curves y
2
= 8x and xy = –1 is
(a) 9x – 3y + 2 = 0 (b) 2x – y + 1 = 0 (c) x – 2y + 8 = 0 (d) x – y + 2 = 0
(27) The lengthof the common chord of the parabolas y
2
= x and x
2
= y is
(a) 1 (b)
2
(c) 4
2
(d) 2
2
(28) The straight line y = a – x touches the parabola x
2
= x – y if a =
(a) –1 (b) 0 (c) 1 (d) 2
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95
(29) If the line x – 1 = 0 is the directrix of the parabola y
2
– kx + 8 = 0 then one of the values
of k is
(a) 4 (b)
1
8
(c)
1
4
(d) 8
(30) If M is the foot of the perpendicular from a point P on a parabola to its directrix and SPM
is an equilateral triangle, where S is the focus, then SP is equal to
(a) 8a (b) 2a (c) 3a (d) 4a
(31) The chord AB of the parabola y
2
= 4ax cuts the axis of the parabola at C. If A = (at
1
2
, 2at
1
),
B = (at
2
2
, 2at
2
) and AB : AC = 3 : 1 then
(a) t
2
= 2t
1
(b) t
1
+ 2t
2
= 0 (c) t
2
+ 2t
1
= 0 (d) t
1
– 2t
2
= 0
(32) Equation of common tangents of y
2
= 4bx and x
2
= 4by is
(a) x + y + b = 0 (b) x – y + b = 0 (c) x – y – b = 0 (d) x + y – b = 0
(33) Angle between the tangents drawn to y
2
= 4x, where it is intersected by the line x – y – 1 = 0
is equal to
(a)
?
2
(b)
?
3
(c)
?
4
(d)
?
6
(34) The angle between the tangents drawn from the point (1, 4) to the parabola y
2
= 4x is
(a)
?
2
(b)
?
3
(c)
?
4
(d)
?
6
(35) The shortest distance between the line x – y + 1 = 0 and the curve x = y
2
is
(a)
3 2
5
(b)
2 3
8
(c)
3 2
8
(d)
2 2
5
(36) Let P be the point (1, 0) and Q a point on the locus y
2
= 8x. The locus of mid–point of PQ is
(a) y
2
+ 4x + 2 = 0 (b) y
2
– 4x + 2 = 0 (c) x
2
– 4y + 2 = 0 (d) x
2
+ 4y + 2 = 0
(37) If tangents to the parabola y
2
= 4ax at the points (at
1
, 2at
1
) and (at
2
2
, 2at
2
) intersect on the
axis of the parabola, then
(a) t
1
t
2
= –1 (b) t
1
t
2
= 1 (c) t
1
= t
2
(d) t
1
+ t
2
= 0
(38) The focus of the parabola x
2
– 8x + 2y + 7 = 0 is
(a)
? ?
F
H
I
K
4
9
2
,
(b)
0
1
2
, ?
F
H
I
K
(c)
9
4,
2
 
 
 
(d) (4, 4)
(39) The point of intersection of the tangents at the ends of the latus rectum of the parabola y
2
= 4x
is
(a) (–1, 0) (b) (1, 0) (c) (0, 0) (d) (0, 1)
Page 4
96
(40) If the line y = 1 – x touches the curve y
2
– y + x = 0, then the point of contact is
(a) (0, 1) (b) (1, 0) (c) (1, 1) (d)
1
2
1
2
,
F
H
I
K
(41) The line y = c is a tangent to the parabola y
2
= 4ax if c is equal to
(a) a (b) 0 (c) 2a (d) None of these
(42) The vertex of the parabola (x – b)
2
= 4b (y – b) is
(a) (b, 0) (b) (0, b) (c) (0, 0) (d) (b, b)
(43) The axis of the parabola 9y
2
– 16x – 12y – 57 = 0 is
(a) y = 0 (b) 16x + 61= 0 (c) 3y – 2 = 0 (d) 3y – 61 = 0
(44) If P(at
2
, 2at) be one end of a focal chord of the parabola y
2
= 4ax, then the length of the chord
is
(a)
a t
t
?
F
H
I
K
1
(b)
a t
t
?
F
H
I
K
1
(c)
a t
t
?
F
H
I
K
1
2
(d)
a t
t
?
F
H
I
K
1
2
(45) The latus rectum of a parabola is a line
(a) through the focus (b) parallel to the directrix
(c) perpendicular to the axis (d) all of these
(46) A tangent to the parabola y
2
= 9x passes through the point (4, 10). Its slope is
(a)
3
4
(b)
9
4
(c)
1
4
(d)
1
3
(47) The line y = mx + 1 is a tangent to the parabola y
2
= 4x if m =
(a) 4 (b) 3 (c) 2 (d) 1
(48) If a chord of the parabola y
2
= 4ax, passing through its focus F meets it in P and Q, then
1
|FP|
1
|FQ|
?
=
(a)
1
a
(b)
2
a
(c)
4
a
(d)
1
2a
(49) The equation of the chord of parabola y
2
= 8x. Which is bisected at the point (2, –3) is
(a) 3x + 4y – 1 = 0 (b) 4x + 3y + 1 = 0 (c) 3x – 4y + 1 = 0 (d) 4x – 3y – 1 = 0
(50) If x + y + 1 = 0 touches the parabola y
2
= ax then a =
(a) 8 (b) 6 (c) 4 (d) 2
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97
(51) If y
1
, y
2
and y
3
are the ordinates of the vertices of a triangle inscribed in the parabola y
2
=
4ax, then its area is
(a)
1
8
1 2 2 3 3 1
a
y y y y y y( ) ( ) ( )? ? ?
(b)
1
4
1 2 2 3 3 1
a
y y y y y y( ) ( ) ( )? ? ?
(c)
1
2
1 2 2 3 3 1
a
y y y y y y( ) ( ) ( )? ? ?
(d)
1
1 2 2 3 3 1
a
y y y y y y( ) ( ) ( )? ? ?
(52) The centre of the ellipse
( ) ( )x y x y? ?
?
?2
9 16
2 2
= 1 is
(a) (1, 1) (b) (0, 0) (c) (0, 1) (d) (1, 0)
(53) Let E be the ellipse
x y
2 2
9 4
1? ? and C be the circle x
2
+ y
2
= 9. Let P and Q be the piont
(1, 2) and (2, 1) respe. Then
(a) P lies inside C but outside E (b) P lies inside both C and E
(c) Q lies outside both C and E (d) Q lies inside C but outside E
(54) The ellipse x
2
+ 4y
2
= 4 is incribed in a rectangle aligned with the co–ordinate axes. Which
in turn is inscribed in an other ellipse that passes through the point (4, 0). Then the equation
of the ellipse is
(a) 4x
2
+ 48y
2
= 48 (b) x
2
+ 16y
2
= 12 (c) x
2
+ 16y
2
= 16 (d) x
2
+ 12y
2
= 16
(55) Chords of an ellipse are drawn through the positive end of the minor axis. Then their mid point
lies on
(a) a circle (b) a parabola (c) an ellipse (d) a hyperbola
(56) The distance from the foci of P(x
1
, y
1
) on the ellipse
x y
2 2
9 25
1? ?
is
(a)
4
5
4
1
? y
(b)
5
4
5
1
? y
(c)
5
4
5
1
? x
(d)
4
4
5
1
? y
(57) If S and S' are two foci of an ellipse 16x
2
+ 25y
2
= 400 and PSQ is a focal chord such that
SP = 16 then S'Q =
(a)
74
9
(b)
54
9
(c)
64
9
(d)
44
9
Page 6
98
(58) Tangents are drawn to the ellipse
x y
2 2
9 5
1? ?
at ends of latus recturm line. The area of
quadrilateral so formed is
(a)
27
4
(b)
27
55
(c) 27 (d)
27
2
(59) Let P be a point on the ellipse
2 2
2 2
1
x y
a b
 
of eccentricity e. If A, A' are the vertices and S,
S' are the foci of an ellipse, then area of ? APA' : area of ? PSS' =
(a) e (b) e
2
(c) e
3
(d)
1
e
(60) A focus of an ellipse is at the origin. The directrix is the line x – 4 = 0 and eccentricity is
1
2
, then the length of semi–major axis is
(a)
5
3
(b)
4
3
(c)
8
3
(d)
2
3
(61) The equation
2 2
1 ; 1
1 1
x y
r
r r
  
 
represents.
(a) a parabola (b) an ellipse (c) a circle (d) None of these
(62) If P(m, n) is a point on an ellipse
x
a
y
b
2
2
2
2
1? ? with foci S and S' and eccentricty e, then area
of ? SPS' is
(a)
ae a m
2 2
?
(b) ae b m
2 2
? (c)
be b m
2 2
?
(d)
be a m
2 2
?
(63) If P(x
1
, y
1
) is a point on an ellipse
x
a
y
b
2
2
2
2
1? ? and it's one focus is S(ae, 0) then PS is equal
to
(a) a + ex
1
(b) a – ex
1
(c) ae + x
1
(d) ae – x
1
(64) If
3
bx + ay = 2ab touches the ellipse
x
a
y
b
2
2
2
2
1? ? then eccentric angle

of point of
contact =
(a)
?
2
(b)
?
3
(c)
?
4
(d)
?
6
Page 7
99
(65) If P is a point on an ellipse 5x
2
+ 4y
2
= 80 whose foci are S and S'. Then PS + PS' =
(a) 4
5
(b) 4 (c) 8 (d) 10
(66) If
x
a
y
b
2
2
2
2
1? ? is an ellipse, then length of it's latus–rectum is
(a)
2
2
b
a
(b)
2
2
a
b
(c) depends on whether a > b or b > a (d)
2
2
a
b
(67) The curve represented by x = 3 (cost + sint); y = 4 (cost – sint) is
(a) circle (b) parabola (c) ellipse (d) hyperbola
(68) The length of the common chord of the ellipse
( ) ( )x y?
?
?
?
1
9
2
4
1
2 2
and the circle
(x – 1)
2
+ (y – 2)
2
= 1
(a)
2
(b)
3
(c) 4 (d) None of these
(69) S and T are the foci of an ellipse and B is an end of the minor axis. If ? STB is an equilateral,
then e =
(a)
1
2
(b)
1
3
(c)
1
4
(d)
1
8
(70) If the line lx + my + n = 0 cuts an ellipse
x
a
y
b
2
2
2
2
1? ? in points whose eccentric angles differ
by
?
2
, then
a l b m
n
2 2 2 2
2
?
=
(a) 1 (b)
3
2
(c) 2 (d)
5
2
(71) Area of the greatest rectangle that can be inscribed in an ellipse
x
a
y
b
2
2
2
2
1? ? is
(a) ab (b) 2ab (c)
a
b
(d)
ab
(72) The equation 2x
2
+ 3y
2
– 8x – 18y + 35 = k represents
(a) parabola if k > 0 (b) circle if k > 0 (c) a point if k = 0 (d) a hyperbola if k > 0
Page 8
100
(73) If
x
a
y
b
? ? 2
touches the ellipse
x
a
y
b
2
2
2
2
1? ? , then its eccentric angle

of the contact
piont is
(a) 0
o
(b) 45
o
(c) 60
o
(d) 90
o
(74) The eccentricity of an ellipse, with its centre at the origin, is
1
2
. If one of the directrices is
x = 4, then equation of an ellipse is
(a) 3x
2
+ 4y
2
= 1 (b) 3x
2
+ 4y
2
= 12 (c) 4x
2
+ 3y
2
= 12 (d) 4x
2
+ 3y
2
= 1
(75) The radius of the circle passing through the foci of the ellipse
x y
2 2
16 9
1? ?
and having its
centre (0, 3) is
(a) 4 (b) 3 (c)
12
(d)
7
2
(76) The equations of the common tangents to the parabola y = x
2
and y = – (x – 2)
2
is
(a) y = 4(x – 1) (b) y = 2 (c) y = –4(x – 1) (d) y = –30x – 50
(77) If e
1
and e
2
be the eccentricities of a hyperbola and its conjugate, then
1 1
1
2
2
2
e e
?
=
(a) 2 (b) 1 (c) 0 (d) 3
(78) A hyperbola, having the transverse axis of length 2 sin? is confocal with the ellipse 3x
2
+
4y
2
= 12. Then its equation is
(a) x
2
cosec
2
? – y
2
sec
2
? = 1 (b) x
2
sec
2
? – y
2
cosec
2
? = 1
(c) x
2
sin
2
? – y
2
cos
2
? = 1 (d) x
2
cos
2
? – y
2
sin
2
? = 1
(79) The locus of a point P(? , ? ) moving under the condition that the line y = ? x + ? is a tangent
to the hyperbola
2 2
2 2
1
x y
a b
 
is
(a) a circle (b) a parabola (c) an ellipse (d) a hyperbola
(80) If (asec? , btan? ) and (asec? , btan? ) are the ends of a focal chord of
2 2
2 2
1
x y
a b
 
then =
tan tan
2 2
 

=
(a)
e
e
?
?
1
1
(b)
1
1
?
?
e
e
(c)
1
1
?
?
e
e
(d)
e
e
?
?
1
1
Page 9
101
(81) If AB is a double ordinates of the hyperbola
2 2
2 2
1
x y
a b
 
such that OAB is an equilateral triangle,
O being the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies.
(a)
1
2
3
? ?e
(b)
e ?
1
3
(c)
e ?
3
2
(d)
e ?
2
3
(82) The value of m for which y = mx + 6 is a tangent to the hyperbola
x y
2 2
100 49
1? ?
is
(a)
17
20
(b)
20
3
(c)
20
17
(d)
3
20
(83) The vertices of the hyperbola 9x
2
– 16y
2
– 36x + 96y – 252 = 0 are
(a) (6, 3), (–6, 3) (b) (–6, 3), (–6, –3) (c) (6, –3), (2, –3) (d) (6, 3),(–2, 3)
(84) Which of the following in independent of ? in the hyperbola
0
2
1
2
2
2
2
? ?
F
H
I
K
? ??
?
? ?
x y
cos sin
?
(a) Vertex (b) Eccentricity (c) Abscissa of foci (d) Directrix
(85) The equation of the tangent to the curve 4x
2
– 9y
2
= 1. Which is parallel to 5x – 4y + 7 = 0
is
(a) 30x – 24y + 17 = 0 (b) 24x – 30y ?
161
= 0
(c) 3x – 24y ?
161
= 0 (d) 24x + 30y ?
161
= 0
(86) Two straight lines pass through the fixed points (? a, 0) and have slopes whose products is
p > 0. Then, the locus of the points of intersection of the lines is
(a) a circle (b) a parabola (c) an ellispe (d) a hyperbola
(87) The equations to the common tangents to the two hyperbola
2 2
2 2
1
x y
a b
 
and are
2 2
2 2
1
y x
a b
 
(a)
y x a b? ? ? ?
2 2
(b)
y x b a? ? ? ?
2 2
(c)
y x a b? ? ? ?
2 2
(d) y = ± x ± (a
2
– b
2
)
(88) If the line
2 6 2x y? ?
touches the hyperbola x
2
– 2y
2
= 4 then the point of contact is
(a) 4 6, ?
c
h
(b)


5, 2 6
(c)
1
2
1
6
,
F
H
I
K
(d) ? 2 6,
c
h
Page 10

Circle and Conic Section (Solved MCQs and Notes)

Circle and Conic Section (Solved MCQs and Notes)

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