ECAT Mathematics
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ECAT Mathematics
Chapter 20: Conic Section
83 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
A conic section is formed by the intersection of a plane with a:
A
Sphere
B
Double right circular cone
C
Cylinder
D
Cube
1
The four types of conics are:
A
Circle, Ellipse, Parabola, Hyperbola
B
Circle, Square, Triangle, Rectangle
C
Circle, Oval, Arc, Segment
D
Sphere, Cone, Cylinder, Cube
2
Standard equation of circle with center (0,0) and radius r:
A
x²+y²=r
B
x²+y²=r²
C
x²/r+y²/r=1
D
x+y=r
3
Standard parabola opening right: y²=4ax has focus at:
A
(−a,0)
B
(a,0)
C
(0,a)
D
(0,0)
4
Directrix of parabola y²=4ax:
A
x=a
B
x=−a
C
y=a
D
y=−a
5
Vertex of parabola y²=4ax:
A
(a,0)
B
(−a,0)
C
(0,0)
D
(a,a)
6
Standard ellipse x²/a²+y²/b²=1 (a>b): foci at:
A
(0,±c) where c²=a²−b²
B
(±c,0) where c²=a²−b²
C
(±a,0)
D
(0,±b)
7
For ellipse x²/a²+y²/b²=1 (a>b), the major axis has length:
A
2b
B
2a
C
a
D
b
8
For hyperbola x²/a²−y²/b²=1, foci at:
A
(±c,0) where c²=a²+b²
B
(0,±c)
C
(±a,0)
D
(±b,0)
9
Asymptotes of x²/a²−y²/b²=1:
A
y=±(b/a)x
B
y=±(a/b)x
C
y=±bx
D
y=±ax
10
Eccentricity e of circle:
A
e>1
B
e=1
C
e=0
D
e<1
11
Eccentricity of parabola:
A
e=0
B
e<1
C
e>1
D
e=1
12
Eccentricity of ellipse:
A
e=0
B
0<e<1
C
e=1
D
e>1
13
Eccentricity of hyperbola:
A
e=0
B
e<1
C
e=1
D
e>1
14
Parabola y²=12x: value of 4a=12, so a=
A
3
B
4
C
12
D
6
15
Focus of y²=12x:
A
(3,0)
B
(0,3)
C
(12,0)
D
(−3,0)
16
Directrix of y²=12x:
A
x=3
B
x=−3
C
y=3
D
y=−3
17
Parabola y²=−8x opens:
A
Right
B
Left
C
Up
D
Down
18
Parabola x²=4y opens:
A
Right
B
Left
C
Up
D
Down
19
Focus of x²=4y:
A
(0,1)
B
(1,0)
C
(0,−1)
D
(4,0)
20
Ellipse x²/16+y²/9=1: a²=16, b²=9, c²=
A
7
B
25
C
√7
D
4
21
Eccentricity of that ellipse:
A
√7/4
B
4/√7
C
7/16
D
4/7
22
Ellipse x²/9+y²/16=1 (a=4>b=3, major axis along y):
A
Foci at (0,±√7)
B
Foci at (±√7,0)
C
Foci at (0,±4)
D
Foci at (±3,0)
23
The latus rectum of parabola y²=4ax has length:
A
4a
B
2a
C
a
D
8a
24
Latus rectum of y²=12x:
A
12
B
3
C
4
D
6
25
The sum of distances from any point on ellipse to both foci equals:
A
2b
B
2c
C
2a
D
a+b
26
The difference of distances from any point on hyperbola to foci equals:
A
2a
B
2b
C
2c
D
a−b
27
Hyperbola x²/9−y²/16=1: a=3, b=4, c=
A
5
B
7
C
√7
D
√25=5
28
Asymptotes of x²/9−y²/16=1:
A
y=±(4/3)x
B
y=±(3/4)x
C
y=±4x
D
y=±(4x)/3
29
The standard form of circle with center (h,k) radius r:
A
(x−h)²+(y−k)²=r
B
(x+h)²+(y+k)²=r²
C
(x−h)²+(y−k)²=r²
D
x²+y²=r²
30
Circle x²+y²+4x−6y+4=0: completing the square gives center:
A
(−2,3)
B
(2,−3)
C
(4,−6)
D
(−4,6)
31
Parabola with vertex at (h,k) opening right:
A
(y−k)²=4a(x−h)
B
(x−h)²=4a(y−k)
C
(y−k)²=4a(x+h)
D
(x+h)²=4a(y+k)
32
Point (5,3) lies on parabola y²=9 (check: 9=9): actually on y²=ax, find a:
A
a=9/25
B
a=5/3
C
a=3
D
a=25/3
33
Focus of parabola x²=−16y:
A
(0,−4)
B
(0,4)
C
(−4,0)
D
(4,0)
34
Directrix of x²=−16y:
A
y=−4
B
y=4
C
x=4
D
x=−4
35
Ellipse: if c=0, then e=0 and ellipse becomes:
A
Parabola
B
Hyperbola
C
Circle (a=b)
D
Degenerate
36
The chord of an ellipse through a focus perpendicular to major axis is called:
A
Vertex
B
Latus rectum
C
Directrix
D
Asymptote
37
Length of latus rectum of ellipse x²/a²+y²/b²=1:
A
2b²/a
B
2a²/b
C
2ab
D
a²/b
38
For x²/25+y²/9=1: length of latus rectum:
A
18/5
B
6
C
18
D
5
39
General second degree: ax²+2hxy+by²+2gx+2fy+c=0. Circle requires:
A
a=b≠0 and h=0
B
a=h=b
C
a=b=h=0
D
h=0 only
40
The parabola y²=4ax has axis of symmetry along:
A
y-axis
B
x-axis
C
y=x
D
y=−x
41
Vertex of ellipse x²/a²+y²/b²=1 on x-axis: major vertices at:
A
(0,±a)
B
(±a,0)
C
(±b,0)
D
(0,±b)
42
The conjugate axis of hyperbola x²/a²−y²/b²=1 lies along:
A
x-axis
B
y-axis
C
y=x
D
y=−x
43
Transverse axis of x²/a²−y²/b²=1 lies along:
A
y-axis
B
x-axis
C
y=x
D
y=−x
44
Rectangular hyperbola xy=c² has eccentricity:
A
1
B
√2
C
2
D
0
45
The eccentricity formula for conic: e=c/a. For circle c=0→e=
A
1
B
0
C
∞
D
2
46
The equation y=x² is a parabola with vertex at:
A
(0,1)
B
(1,0)
C
(0,0)
D
(1,1)
47
For y=x²: axis of symmetry is:
A
x-axis
B
y-axis
C
y=x
D
No axis
48
Parabola opening upward: vertex minimum. Parabola opening downward: vertex is:
A
Minimum
B
Maximum
C
Inflection point
D
Asymptote
49
The equation x²/4+y²/4=1 represents:
A
Ellipse
B
Hyperbola
C
Parabola
D
Circle (r=2)
50
Circle x²+y²=0 is:
A
Empty set
B
Just the origin (0,0)
C
Entire plane
D
Not a conic
51
Degenerate conics include:
A
Circles and ellipses only
B
Points, lines, pairs of lines
C
Parabolas only
D
Hyperbolas only
52
For ellipse with a=5, b=3, c²=?
A
34
B
16
C
25
D
9
53
For that ellipse, foci at:
A
(±4,0)
B
(±3,0)
C
(±5,0)
D
(±2,0)
54
Parabola y²=4x+4y: rearrange as y²−4y=4x→(y−2)²=4x+4→(y−2)²=4(x+1). Vertex:
A
(−1,2)
B
(1,−2)
C
(0,2)
D
(−1,0)
55
Focus of (y−2)²=4(x+1): a=1, focus at (h+a,k)=
A
(0,2)
B
(−2,2)
C
(1,2)
D
(−1,3)
56
The eccentricity of rectangular hyperbola (x=a is asymptote):
A
1
B
√2
C
2
D
0
57
The ellipse x²/9+y²/4=1: length of major axis=
A
4
B
3
C
6
D
√5
58
For that ellipse, length of minor axis:
A
4
B
2
C
6
D
√5
59
Vertex of parabola (x+2)²=8(y−3):
A
(−2,3)
B
(2,−3)
C
(−2,−3)
D
(2,3)
60
Which of the following is an equation of a hyperbola?
A
x²+y²=9
B
x²/4+y²/9=1
C
x²/4−y²/9=1
D
x²=8y
61
The equation xy=6 represents:
A
Circle
B
Ellipse
C
Rectangular hyperbola
D
Parabola
62
The axis of symmetry of x²=8y:
A
x-axis
B
y-axis
C
y=x
D
y=−x
63
For x²=8y: focus at:
A
(0,2)
B
(2,0)
C
(0,−2)
D
(8,0)
64
For x²=8y: directrix:
A
y=−2
B
y=2
C
x=2
D
x=−2
65
The sum PF₁+PF₂ for point on ellipse x²/25+y²/16=1:
A
10
B
16
C
25
D
8
66
Chord of contact of tangents from (x₁,y₁) to circle x²+y²=r² is:
A
xx₁+yy₁=r²
B
x+y=r
C
x₁y+y₁x=r
D
xx₁−yy₁=r²
67
The locus of a point that moves so that its distance from fixed point F equals its distance from fixed line d is:
A
Circle
B
Ellipse
C
Parabola
D
Hyperbola
68
The locus where sum of distances to two fixed points is constant is:
A
Circle
B
Ellipse
C
Parabola
D
Hyperbola
69
The locus where |difference| of distances to two fixed points is constant is:
A
Circle
B
Ellipse
C
Parabola
D
Hyperbola
70
Hyperbola x²/16−y²/9=1: a=4, b=3, c=
A
5
B
7
C
√7
D
13
71
For that hyperbola, eccentricity e=
A
5/4
B
4/5
C
5/3
D
4/3
72
For that hyperbola, asymptotes:
A
y=±(3/4)x
B
y=±(4/3)x
C
y=±(5/4)x
D
y=±3x
73
Ellipse with e=1/2 and a=6 has c=
A
3
B
6
C
1/12
D
12
74
For that ellipse, b²=a²−c²=
A
27
B
9
C
36
D
33
75
Equation of that ellipse (major axis along x): x²/36+y²/27=1. At x=6: y=
A
0
B
√27=3√3
C
1
D
27
76
The parabola y=ax²+bx+c has vertex x-coordinate:
A
−b/2a
B
b/2a
C
a/2b
D
−a/2b
77
For y=x²−4x+3: vertex at x=−(−4)/(2)=2; y=4−8+3=
A
−1
B
1
C
0
D
−2
78
So vertex is (2,−1). The parabola opens:
A
Down
B
Up (a=1>0)
C
Left
D
Right
79
x-intercepts of y=x²−4x+3: set y=0: (x−1)(x−3)=0: x=
A
1 and 3
B
2 and 2
C
−1 and 3
D
1 and −3
80
The circle x²+y²−6x+8y=0 center and radius:
A
Center(3,−4), r=5
B
Center(−3,4), r=5
C
Center(3,4), r=5
D
Center(3,−4), r=25
81
The point (5,0) relative to circle x²+y²=16:
A
Inside
B
On the circle
C
Outside
D
At center
82
The tangent to circle x²+y²=r² at point (x₁,y₁) is:
A
x+y=r
B
xx₁+yy₁=r²
C
x/x₁+y/y₁=1
D
xx₁−yy₁=r
0
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