Coordinate geometry questions and solutions

Question1 : A(1,1) , B(5,5) and C(8,2) are the vertices of a triangle. Find the perimeter of the triangle ABC.
Option 1 : 9√2 units
Option 2 : 10√2 units
Option 3 : 12√2 units
Option 4 : 14√2 units
Explanation : Distance between A and B
= √[(x2-x1)²+(y2-y1)²]
= √[(5-1)²+(5-1)²]
= √(16+16)
= 4√2 units
Distance between B and C
= √[(8-5)²+(2-5)²]
= √[9+9]
= √18
= 3√2 units
Distance between C and A
= √[(8-1)²+(2-1)²]
= √[49+1]
= √50
= 5√2 units
Perimeter of the triangle
= 4√2+3√2+5√2
= 12√2 units

Question2 : What is the equation of the line that crosses the x-axis at (3,0) and the y-axis at (0,4)?
Option 1 : 3y-4x=12
Option 2 : 4y-3x=6
Option 3 : 3x-4y=6
Option 4 : 4x+3y=12
Explanation : The equation is the line is of the form
y = mx+c
Slope of the line
= (y2-y1)/(x2-x1)
= 4/-3
= -4/3
m = -4/3
To find c, substitute the point (0,4) in the equation of the line.
y = mx+c
4 = 0+c
c = 4
Hence, the equation of the line is
y = (-4/3)x+4
3y = -4x+12
4x+3y = 12

Question3 : What is the slope of the line that passes through (4,5) and is parallel to the line that passes through (-4,5) and (4,-6)?
Option 1 : -11/8
Option 2 : 11/8
Option 3 : -8/11
Option 4 : 8/11
Explanation : Parallel lines have same slopes.
Slope of the line passing through (-4,5) and (4,-6)
= (y2-y1)/(x2-x1)
= (-6-5)/(4-(-4))
= -11/8

Question4 : Find the slope of the line that passes through (p,q) and (p+10,1/q).
Option 1 : p+(10/q²)
Option 2 : 1/pq
Option 3 : (1-q²)/10q
Option 4 : (p-1)/q
Explanation : Slope
= (y2-y1)/(x2-x1)
= [(1/q)-q]/(p+10-p)
= (1-q²)/10q

Question5 : What is the slope of a line which is perpendicular to y-axis?
Option 1 : 1
Option 2 : 0
Option 3 : 1/2
Option 4 : -1
Explanation : A line perpendicular to y axis will be horizontal and
the slope of a horizontal line is always zero.

Question6 : The slope of the line passing through two points (a,a) and (4a,b) is -1. Find b in terms of a.
Option 1 : -2a
Option 2 : -4a
Option 3 : a/4
Option 4 : -a/2
Explanation : Slope of a line passing through two points
= (y2-y1)/(x2-x1)
= (b-a)/(4a-a)
= (b-a)/3a
(b-a)/3a = -1
b-a = -3a
b = -2a

Question 7: If P(2,8) and Q(5,4) are the endpoints of one side of a square PQRS, what is the area of the square in square units?
Option 1 : 2√5
Option 2 : 10
Option 3 : 7
Option 4 : 25
Explanation : The length of a side of the square is equal to the distance between the points P and Q.
Distance between P and Q
= √[(x2-x1)²+(y2-y1)²]
= √[(5-2)²+(4-8)²]
= √(9+16)
= √25
= 5 units
Area of the square
= 5²
= 25 sq. units

Question 8: Find the distance between the points A(3,11) and B(8,-4).
Option 1 : 5√2 units
Option 2 : 25√10 units
Option 3 : 5√10 units
Option 4 : 2√5 units
Explanation : Distance between two points (x1,y1) and (x2,y2) is
d = √[(x2-x1)²+(y2-y1)²]
= √[(8-3)²+(-4-11)²]
= √[5²+(-15)²]
= √(25+225)
= √250
= 5√10 units

Question9 : (x,y) is a point below x-axis and to the right of y-axis. Which of the following is true?
Option 1 : x>y
Option 2 : x<y
Option 3 : x=y
Option 4 : y>0
Explanation : Below x-axis.
So, y is negative.
Right of y-axis.
So, x is positive
x>y

Question10 : Find the slope of the line 3y-5x=2.
Option 1 : 2/3
Option 2 : 2/5
Option 3 : 3/5
Option 4 : 5/3
Explanation : 3y - 5x = 2
Change it to the form
y=mx+c
3y = 2+5x
y = (5/3)x+(2/3)
m = 5/3

Question11 : Find the point at which the line y=3x+8 touches the y-axis.
Option 1 : (0,8)
Option 2 : (8,0)
Option 3 : (3,0)
Option 4 : (0,3)
Explanation : The x coordinate of the point will be zero since it lies on the y axis.
y coordinate
= 3*(0)+8
= 8
The line touches the y axis at (0,8)

Question12 : Find the equation of the line passing between (4,0) and (0,6).
Option 1 : 4x+3y=24
Option 2 : 3x+4y=24
Option 3 : 3x+2y=12
Option 4 : 2x+3y=12
Explanation : The equation of a line passing through two points (x1,y1) and (x2,y2) is
(y-y1)/(y2-y1) = (x-x1)/(x2-x1)
(y-0)/6 = (x-4)/-4
-4y = 6x-24
6x+4y = 24
3x+2y=12

Question13 : A line l passes through two points (3,5) and (1,-3). Another line m is perpendicular to l. Find the slope of m.
Option 1 : 4
Option 2 : -4
Option 3 : 1/4
Option 4 : -1/4
Explanation : Slope of line l
= (y2-y1)/(x2-x1)
= (-3-5)/(1-3)
= 4
Slope of a perpendicular line is the negative reciprocal of the slope of this line.
Slope of m
= -1/4

Question14 : We have two lines l1 and l2.
l1 is parallel to x-axis.
l2 passes between points (1,1) and (2,2)
Which line has the greatest slope?
Option 1 : l1
Option 2 : l2
Option 3 : Both are equal
Option 4 : Data insufficient
Explanation : l1 is a horizontal line.
Slope of a horizontal line=0
Slope of l2
= (y2-y1)/(x2-x1)
= 1
Slope of l2>Slope of l1

Question15 : P(-2,4) and Q(4,-2) are the end points of the diameter of a circle. What are the coordinates of the center of the circle?
Option 1 : (-2,-2)
Option 2 : (2,2)
Option 3 : (-1,-1)
Option 4 : (1,1)
Explanation : The midpoint of the line segment joining two points (x1,y1) and (x2,y2) is given by
((x1+x2)/2, (y1+y2)/2)
The midpoint of the diameter is the center of the circle.
((-2+4)/2, (4-2)/2) = (1,1)

Question16 : Find the distance between the points (-1,2) and (5,10).
Option 1 : 6
Option 2 : 8
Option 3 : 10
Option 4 : 12
Explanation : Distance between two points (x1,y1) and (x2,y2) is
d = √[(x2-x1)²+(y2-y1)²]
= √[(5-(-1))²+(10-2)²]
= √[6²+8²]
= √(36+64)
= √100
= 10 units

Question 17: Find the y-coordinate of (4,-6) when reflected over X-axis.
Option 1 : 4
Option 2 : -4
Option 3 : 6
Option 4 : -6
Explanation : When a point is reflected over X axis, the x co-ordinate remains the same.
The y co-ordinate changes sign.
y co-ordinate = 6

Question18 : A line passes through the points (6,7) and (4,a). If its slope is 1/2, find a.
Option 1 : 5
Option 2 : 4.5
Option 3 : 6
Option 4 : 7.5
Explanation : The slope of a line passing through two points (x1,x2) and (y1,y2) is given by
m = (y2-y1)/(x2-x1)
1/2 = (a-7)/(4-6)
(4-6) = 2(a-7)
-2 = 2a-14
2a=12
a=6

Question19 : If f(x) = x² - 25, how many times does the graph of f(x) cross the x-axis?
Option 1 : 0
Option 2 : 1
Option 3 : 2
Option 4 : 5
Explanation : f(x) = x²-25
= (x+5)(x-5)
The graph of f(x) crosses the x axis when f(x)=0
(x+5)(x-5) = 0
x=-5, x=5
It crosses the x-axis at x=-5 and x=5.

Question20 : If the equation of a line is given by y = 4x+5, then which of the following equations represents the equation of a line parallel to the given line?
Option 1 : y=(1/4)x+5
Option 2 : y=2x+5
Option 3 : y=4x-3
Option 4 : y=-4x+5
Explanation : If the equation of a line is of the form
y = mx+c
m is the slope.
Hence, the slope of the line y=4x+5 is
m=4
Parallel lines have same slopes.
Hence, from the given options,
only option (C) has slope m=4.

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