Problem Solving

If centre is (0,0) and radius is 5, then for x and y we need to test from -5 to +5 including 0.
But if we just try for x from 0 to 5. We know that potential combos are (5,0); (3,4).
For each number if we have positive and negative except of course 0 we have, 8 possibilities for (3,4) combination i.e. (3,4); (-3,4)...and so on including the reversal when x and y are interchanged.
For (5,0) we have 4 combinations i.e. (5,0); (-5,0) and so on including the reversal when x and y are interchanged.

So, total is 8+4 = 12 possibilities.

the solution is the no of solutions for the equation x^2+y^2 = 25.
The integral solutions are (0,+/-5) (+/-5,0) (+/-4,+/-3), (+/-3,+/-4). totaling 12 points.

(3,4,5) is a pythagorean triplet. Hence 4 point with all positive negative combination of (3,4) and (4,3) are also on the circle.
12 points are integral.

Can someone tell me why the Pythagorean Theorem was used and how they came up with the points?

Rastis wrote:Can someone tell me why the Pythagorean Theorem was used and how they came up with the points?

Well there are 4 possibilities when the radius will lie on the axis(0,-5),(0,5),(5,0)(-5,0).
Now we need to find other points in the co-ordinate planes which have integer values.
You can remember the equation of the circle as x^2+y^2=r^2.Where x and y are the co-ordinates of the point,the circle will pass through and r is the radius,or the distance between the origin and the point(x,y).If you look closely this equation is also the pythagorous equation.
Another way to look at it is suppose you drop a perpendicular from point (x,y) on the x axis to form a triangle with the origin,the base of the perpendicular(0,x) and point x,y.This triangle is a right angled triangle.

Thank you for that clarification, shubhamkumar. I didn't know the equation of a circle so that was one issue. However, I still do not know how the other points were found that add up to 12.

(3,4,5) are pythagorean triplets
Hence the integer points are (+/-3,+/-4), (+/-4,+/-3) (+/-5,0) and (0,+/-5) total 12 points

but arent' there an infinite number of points on a circle? I am confused as to why those points are the only ones that are integers.

the center of the circle is origin(0,0).
Any point on the given circle satisfies the equation x^2+y^2 = 5^2(5 being the radius).So all the infinite points on the circle will satisfy this equation.
Now you have to identify the points which will be integer.Start from X=0 so this will give 0^2+y^2=25
which gives Y = +5 and -5 so two points are 0,5 and 0,-5.
Again put x=1 then this will not give integer value for y.
So test X=3 and X=-3 which will give Y=+4 and Y=-4
like wise test for X=+4 and X=-4 and x=+5 and X=-5

So you will get a total of 12 points only having both x and y as integers.

The points will be the solution for the following eq

x^2+y^2=25
0,+-5,+-4,+-3

we have 12 points
sp 12 is the ans

C) 12

x^2+y^2=25

IMO also C.

Rastis wrote:but arent' there an infinite number of points on a circle? I am confused as to why those points are the only ones that are integers.

Integers must be whole numbers and in this case there are only 12 options. Think of just one section of this circle and it makes the problem a bit easier. If you think of just one quarter section then multiply the answer by 4 it is easier.

You know the radius is 5 so at the y axis you have (0,5) and the x axis you have (5,0). If triangles are comfortable for you, think of how many different right (90 degree) triangles you can make with integer base and heights with a hypotenuse length of 5.

The common 3-4-5 triangle is the only option. It can be flipped so that the 3 refers to the x or y axis giving us two additional options for the 1/4 section.

Take all of the option so far (4) and take away one of the options that will be replicated (I took away the y-axis (0,5) and multiply the 3 by 4 sections of the circle for 12 total.

That is how I thought of it! Hope it helps!

Here is another Question on similar lines for you to practice...

Answer: Option D

Saying pythagorus theorem is wrong way.

we know the distance of x,y to 0,0 is 5
so
x^2+ y^2 = 5^2

put x as 0,2,3,5 & get y as 5,3,2,0

total of 12 points.

C