Solutions to questions on Coordinate Geometry

Have given the solutions to questions on Cogeo

1. Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

Any point on the line x/3 + y/5 =1 will have the shortest overall distance. However, we need to have integral coordinates. So, we need to find points with integral coordinates as close as possible to the line 5x + 3y =15.

Substitute x =1, we get y = 2 or 3
Substitute x = 2, we get y = 1 or 2


Sum of distances for ( 1, 2) = sqrt(8) + sqrt (10)
Sum of distances for ( 1, 3) = sqrt(13) + sqrt (5)
Sum of distances for ( 2, 1) = sqrt(2) + sqrt (20)
Sum of distances for ( 2, 2) = sqrt(5) + sqrt (13)

sqrt(5) + sqrt (13) is the shortest distance. Sum of abscissa + ordinate = 4

2. Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2?

Line 3x + 4y =12 cuts the x-axis at (4, 0) and y axis at (0, 3)

The region in the first quadrant satisfying the condition 3x + 4y < 12 forms aright triangle with sides 3, 4 and 5. Area of this triangle = 6 sq units.

The lines x = 2 and 3x + 4y = 12 intersect at (2, 1.5). So, the region r > 2, 3x + 4y < 12 also forms a right triangle. This right triangle has base sides 2, 1.5. Area of this triangle = 1.5

Probability of the point lying in said region = 1.5/6 = 1/4

3. Region Q is defined by the equation 2x + y < 40. How many points (r, s) exist such that r is a natural number and s is a multiple of r?

When r = 1, s can take 37 values [37/1]
When r = 2, s can take 17 values [35/2]
When r = 3, s can take 11 values [33/3]
When r = 4, s can take 7 values [31/4]
When r = 5, s can take 5 values [29/5]
When r = 6, s can take 4 values [27/6]
When r = 7, s can take 3 values [25/7]
When r = 8, s can take 2 values [23/8]
When r = 9, s can take 2 values [21/9]
When r = 10, s can take 1 values [19/10]
When r = 11, 12, 13 s can take 1 values one value each

Totally, there are 92 values possible