# Coordinate geometry questions and solutions

**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

Answer : C

**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

Answer : D

**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

= (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

c = 4

Hence, the equation of the line is

y = (-4/3)x+4

3y = -4x+12

4x+3y = 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

Answer : A

**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

= (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

Answer : C

**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

Answer : B

**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

Answer : A

**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

Answer : D

**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

= √[(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

Answer : C

**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

Answer : A

**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

Answer : D

**Explanation :**3y - 5x = 2

Change it to the form

y=mx+c

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)

Answer : A

**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)

= 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

Answer : C

**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

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

Answer : D

**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

= -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

Answer : B

**Explanation :**l1 is a horizontal line.

Slope of a horizontal line=0

Slope of l2

= (y2-y1)/(x2-x1)

= 1

= (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)

Answer : D

**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

Answer : C

**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

Answer : C

**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

Answer : C

**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

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

Answer : C

**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

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

Answer : C

**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

m=4

Parallel lines have same slopes.

Hence, from the given options,

only option (C) has slope m=4.

only option (C) has slope m=4.

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