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Problem 20

# The number of points with integral co-ordinates that are interior to the circle $$x^{2}+y^{2}=16$$ is : (a) 43 (b) 45 (c) 47 (d) 49

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The number of points with integral coordinates that are interior to the circle $$x^{2}+y^{2}=16$$ is 145.
See the step by step solution

## Step 1: Understanding the circle's co-ordinates

We first need to understand that for a point (x, y) to be an interior point of the circle $$x^{2}+y^{2}=16$$, the sum of the squares of x and y coordinates should be less than the square of the radius (which is $$4^{2}$$ in our case). Thus, we have $$x^{2} + y^{2} < 16$$.

## Step 2: Counting the points

We now need to count the number of integral solutions to $$x^{2}+y^{2}<16$$. Remember, x and y can take positive as well as negative values. On manual counting, we get: For x=0, the permissible y values are -3, -2, -1, 0, 1, 2, 3 (7 values). For x=1 or -1, y values can be -3,-2,-1,0,1,2,3 (7 values); this makes 14 values for y for these two x's. For x=2,-2, y values can be -2,-1,0,1,2 (5 values); this makes 10 values for these two x's. For x=-3,3, y values can be -1,0,1 (3 values); this makes 6 values for these two x's. We take the sum of total values for y to get our answer.

## Step 3: Adding up all counted points

We now add up all the permissible values of the y coordinate for each x coordinate. We have 7+14+10+6 = 37 for points in each quadrant. Because we have 4 quadrants in total, we multiply this count by 4, which gives us 37 * 4 = 148. But, this includes the origin (0,0) four times (once in each quadrant), so we subtract 3 to avoid over-counting to get the final result.

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