ECAT Mathematics
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ECAT Mathematics
Chapter 18: Analytic Geometry
82 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
Distance between (x₁,y₁) and (x₂,y₂):
A
√((x₂−x₁)+(y₂−y₁))
B
√((x₂−x₁)²+(y₂−y₁)²)
C
(x₂−x₁)²+(y₂−y₁)²
D
|x₂−x₁|+|y₂−y₁|
1
Midpoint of (x₁,y₁) and (x₂,y₂):
A
((x₁+x₂),(y₁+y₂))
B
((x₁+x₂)/2, (y₁+y₂)/2)
C
((x₁−x₂)/2, (y₁−y₂)/2)
D
((x₂−x₁), (y₂−y₁))
2
Slope of line through (x₁,y₁) and (x₂,y₂):
A
(y₂−y₁)/(x₂−x₁)
B
(x₂−x₁)/(y₂−y₁)
C
(y₂+y₁)/(x₂+x₁)
D
(x₂−x₁)−(y₂−y₁)
3
Equation of line with slope m and y-intercept b:
A
y=mx
B
y=mx+b
C
y=x+b
D
y=bx+m
4
Equation of line through (x₁,y₁) with slope m:
A
y−y₁=m(x−x₁)
B
y=mx
C
y₁=mx₁+b
D
y+y₁=m(x+x₁)
5
Two lines are parallel if their slopes:
A
Are equal
B
Are negative reciprocals
C
Sum to 1
D
Are both 0
6
Two lines are perpendicular if their slopes:
A
Are equal
B
Multiply to give −1
C
Sum to 0
D
Are reciprocals
7
The slope of a horizontal line is:
A
∞
B
0
C
1
D
Undefined
8
The slope of a vertical line is:
A
0
B
1
C
∞ (undefined)
D
Negative
9
Distance from point (x₀,y₀) to line ax+by+c=0:
A
|ax₀+by₀+c|/√(a²+b²)
B
ax₀+by₀+c
C
(ax₀+by₀+c)/(a+b)
D
|ax₀+by₀|/(a²+b²)
10
Equation of line with intercepts a and b: x/a+y/b=
A
ab
B
0
C
1
D
a+b
11
The x-intercept of line 3x+4y=12:
A
3
B
4
C
12
D
6
12
The y-intercept of line 3x+4y=12:
A
3
B
4
C
12
D
8
13
Slope of 2x−3y+6=0:
A
3/2
B
2/3
C
−2/3
D
−3/2
14
Equation of line through (0,0) and (3,4):
A
3x−4y=0
B
4x−3y=0
C
4x+3y=0
D
y=3/4x
15
Two lines ax+by+c=0 and px+qy+r=0 are parallel if:
A
a/p=b/q≠c/r
B
a/p=b/q=c/r
C
aq=bp
D
ap+bq=0
16
Point dividing AB with A(1,2), B(5,6) in ratio 1:1:
A
(3,4)
B
(2,4)
C
(4,4)
D
(3,3)
17
Section formula: point dividing (x₁,y₁) and (x₂,y₂) in ratio m:n internally:
A
((mx₁+nx₂)/(m+n),(my₁+ny₂)/(m+n))
B
((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n))
C
((mx₁−nx₂)/(m+n),...)
D
(m×x₁,n×y₂)
18
Circle with center (h,k) and radius r: equation:
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
19
Circle x²+y²=25 has center and radius:
A
Center(0,0), r=25
B
Center(0,0), r=5
C
Center(5,5), r=5
D
Center(0,0), r=√5
20
Circle (x−3)²+(y+2)²=16 has center:
A
(3,2)
B
(−3,2)
C
(3,−2)
D
(−3,−2)
21
Circle x²+y²+2gx+2fy+c=0 has center:
A
(g,f)
B
(−g,−f)
C
(2g,2f)
D
(g/2,f/2)
22
Radius of x²+y²+2gx+2fy+c=0:
A
g²+f²−c
B
√(g²+f²−c)
C
g+f
D
g²−f²+c
23
Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃):
A
(1/2)|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|
B
(1/2)(x₁y₂+x₂y₃)
C
|x₁y₂−x₂y₁|
D
(1/2)base×height only
24
Area of triangle (0,0),(4,0),(0,3):
A
6
B
12
C
7
D
5
25
Three points are collinear if area of triangle formed = ?
A
1
B
0
C
∞
D
Area is never 0
26
Perpendicular bisector of segment AB passes through:
A
A and B
B
Midpoint of AB, perpendicular to AB
C
A only
D
An endpoint
27
Line 4x+3y=12 intersects x-axis at:
A
(3,0)
B
(0,4)
C
(4,0)
D
(0,3)
28
Line 4x+3y=12 intersects y-axis at:
A
(0,3)
B
(0,4)
C
(3,0)
D
(4,0)
29
Slope of perpendicular to line with slope 2/3:
A
2/3
B
3/2
C
−3/2
D
−2/3
30
The angle θ between two lines with slopes m₁ and m₂:
A
tanθ=|m₁−m₂|/(1+m₁m₂)
B
tanθ=(m₁+m₂)/(1−m₁m₂)
C
tanθ=m₁m₂
D
sinθ=m₁−m₂
31
Locus of points equidistant from (3,0) and (−3,0) is:
A
x=3
B
y=0 (x-axis)
C
x=0 (y-axis)
D
y=x
32
Distance from (0,0) to line 3x+4y+10=0:
A
10/5=2
B
10/25
C
2
D
√25=5
33
Distance between parallel lines 3x+4y+5=0 and 3x+4y+10=0:
A
5/5=1
B
5
C
1
D
10/5=2
34
Equation of x-axis:
A
x=0
B
y=0
C
y=x
D
x+y=0
35
Equation of y-axis:
A
x=0
B
y=0
C
y=x
D
x+y=0
36
The gradient (slope) of y=3x+5:
A
5
B
3
C
3/5
D
1/3
37
Point on line y=2x+1 when x=3:
A
y=7
B
y=5
C
y=6
D
y=3
38
Do lines y=2x+1 and y=2x−3 intersect?
A
Yes, at one point
B
No (parallel, same slope)
C
Yes, infinitely many
D
At x=0
39
Lines y=x+2 and y=−x+4 intersect at:
A
(1,3)
B
(2,4)
C
(0,2)
D
(3,1)
40
A quadrilateral has vertices at (0,0),(4,0),(4,3),(0,3). It is a:
A
Square
B
Rectangle
C
Rhombus
D
Trapezoid
41
Area of that quadrilateral:
A
12
B
16
C
9
D
7
42
Midpoint of (−2,4) and (6,−2):
A
(2,1)
B
(4,2)
C
(2,1)
D
(−2,1)
43
Distance from (3,4) to origin:
A
5
B
7
C
12
D
√(9+16)=5
44
Angle made by line y=x with positive x-axis:
A
30°
B
45°
C
60°
D
90°
45
Equation of line parallel to x-axis through (3,5):
A
x=3
B
y=5
C
y=x+3
D
y=3x+5
46
Equation of line parallel to y-axis through (3,5):
A
x=3
B
y=5
C
x=5
D
y=3
47
The centroid of triangle (x₁,y₁),(x₂,y₂),(x₃,y₃):
A
((x₁+x₂+x₃)/3,(y₁+y₂+y₃)/3)
B
((x₁+x₂+x₃)/2,...)
C
(x₁x₂x₃,y₁y₂y₃)
D
Midpoint of any side
48
Centroid of (0,0),(6,0),(0,4):
A
(2,4/3)
B
(3,2)
C
(2,2)
D
(6,4)
49
The general second-degree equation ax²+2hxy+by²+2gx+2fy+c=0 represents a conic. If a=b and h=0, it represents:
A
Parabola
B
Hyperbola
C
Ellipse
D
Circle
50
The condition for parabola (from general form):
A
h²=ab
B
h²>ab
C
h²<ab
D
a=b
51
For hyperbola:
A
h²=ab
B
h²<ab
C
h²>ab
D
a=−b
52
For ellipse:
A
h²=ab
B
h²<ab
C
h²>ab
D
a=b
53
Distance from origin to line ax+by+c=0:
A
|c|/√(a²+b²)
B
(a+b+c)/√(a²+b²)
C
c/(a+b)
D
|a+b|/c
54
The line through (x₁,y₁) perpendicular to ax+by+c=0 has equation:
A
ax+by=ax₁+by₁ (wait, perpendicular uses slope −a/b... which gives bx−ay=bx₁−ay₁)
B
ax+by=c
C
bx−ay=bx₁−ay₁
D
a(x−x₁)+b(y−y₁)=0
55
Length of tangent from external point (x₁,y₁) to circle x²+y²=r²:
A
√(x₁²+y₁²−r²)
B
√(r²−x₁²−y₁²)
C
x₁²+y₁²
D
r
56
Circle x²+y²−4x+2y=0: center and radius:
A
Center(2,−1), r=√5
B
Center(−2,1), r=√5
C
Center(2,1), r=5
D
Center(4,−2), r=5
57
The line y=mx+c is tangent to x²+y²=r² if:
A
c=mr
B
c²=r²(1+m²)
C
c²=r²m²
D
mr=c²
58
Collinear points (2,4),(4,8),(k,12) require k=?
A
6
B
8
C
5
D
10
59
Area of triangle with base b and height h:
A
bh
B
bh/2
C
2bh
D
b+h
60
If A(1,2) and B(5,8), slope of AB:
A
(8−2)/(5−1)=6/4=3/2
B
6/5
C
1/2
D
3/4
61
The foot of perpendicular from (a,b) to x-axis:
A
(a,b)
B
(0,b)
C
(a,0)
D
(0,0)
62
The reflection of (3,4) in x-axis:
A
(3,4)
B
(−3,4)
C
(3,−4)
D
(−3,−4)
63
The reflection of (3,4) in y-axis:
A
(3,4)
B
(−3,4)
C
(3,−4)
D
(−3,−4)
64
The reflection of (3,4) in origin:
A
(3,4)
B
(−3,4)
C
(3,−4)
D
(−3,−4)
65
Line x+y=5 has slope:
A
1
B
−1
C
5
D
1/5
66
Line x−2y=6: slope:
A
1/2
B
−1/2
C
2
D
−2
67
Are (1,2),(3,6),(5,10) collinear?
A
No
B
Yes (slope=2 throughout)
C
Cannot determine
D
Only two are collinear
68
Internal division of (0,0),(6,4) in ratio 1:2:
A
(2,4/3)
B
(3,2)
C
(4,8/3)
D
(1,2/3)
69
External division of (1,2),(7,8) in ratio 1:3:
A
(−2,−1)
B
(4,5)
C
(5,6)
D
(7,8)
70
Shortest distance from line to a point is measured:
A
Along the line
B
Perpendicular to the line
C
At 45°
D
Along x-axis
71
Intersection of x+y=5 and 2x−y=1:
A
(2,3)
B
(3,2)
C
(1,4)
D
(4,1)
72
Equation of perpendicular bisector of (0,0) and (4,6):
A
2x+3y=13
B
x=2
C
3x+2y=13
D
2x+3y=26
73
The locus of points equidistant from (1,0) and (7,0) is:
A
x=3
B
x=4
C
y=4
D
x=4 (perpendicular bisector)
74
Slope of line making 135° with x-axis:
A
1
B
−1
C
tan135°=−1
D
Both B and C
75
Equation of line through (2,3) parallel to y=3x+1:
A
y=3x−3
B
y=3x+3
C
y=x+3
D
y=−x/3+3
76
Equation of line through (1,2) perpendicular to 2x+3y=6:
A
3x−2y=−1
B
2x+3y=8
C
3x−2y=1
D
2x−3y=4
77
The point of concurrence of medians of a triangle is called:
A
Incenter
B
Circumcenter
C
Centroid
D
Orthocenter
78
The circumcenter is equidistant from all:
A
Sides
B
Midpoints of sides
C
Vertices
D
Altitudes
79
If distance between (a,2) and (3,4) is √8: a=?
A
1
B
5
C
3
D
7
80
Angle bisectors of coordinate axes have equations:
A
x=0 or y=0
B
y=x or y=−x
C
y=1 or y=−1
D
x=1 or x=−1
81
The locus of midpoints of chords of circle x²+y²=r² from point (a,b):
A
(x−a/2)²+(y−b/2)²=(r/2)²
B
x²+y²=a²+b²
C
A circle with center (a/2,b/2)
D
Line through origin
0
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