Question

In the xy-plane, C and D are circles centered at the origin with radii p 17 and p 5, respectively. Quantity A: The number of points (a; b) on circle C where both a and b are integers Quantity B: The number of points (a; b) on circle D where both a and b are integers A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Answer 1

The answer is D

Explanation:

Circle C covers circle D therefore all the dots on the circle D also on circle C. It makes that Quantity B is most likely greater that Quantity A but there is no certainty on this comparison. Those quantity may be equal but we can not say they are exactly equal. Thus, the relationship cannot be determined from the information given.

Answer 2

Quantity A is greater

Explanation:

Well, to solve this we'll do it by parts.

Quantity A) 809 coordinate points

Since it's centered at the origin we can write the circle equation as

`x^(2) +y^(2) =17^(2)</p><p>We can rewrite it as:</p><p>[tex]x^(2) +y^(2)=289`

For now, we set apart the following points (0,17) (0,-17), (17,0) and (-17,0) because those points already belong to the circumference and circle.

Now we're going to enlist the cartesian points. The parameter is the Radius. No point can be outside the circle, i.e. with a distance longer than 289 units.

Doing it with the Quadrant I. We'll enlist them. And test by squaring

(x,y) x²+y²≤289

(0,0) Origin

(1,1); ...(1,16) (1,16)∈ for 1²+16² ≤289

(1,1); ...(1,16) = 16 points

(2,1)....(2,16) =16 points (7,1)...(7,15) =15 points (12,1)...(12,12)=12 points

(3,1)....(3,16) =16 points (8,1)....(8,15) =15 points (13,1)...(13,10)= 10 points

(4,1)....(4,16) =16 points (9,1)...(9,14) =14 points (14,1)...(14,9)= 9 points

(5,1)....(5,15) =15 points (10,1)...(10,13) =13 points (15,1)...(15,8) =8 points

(6,1)...(6,15) =15 points (11,1)...(11,12)= 12 points (16,1)...(16,5) =5 points

So

16+16+16+15+15+15+15+14+13+12+12+10+9+8+5=

48+60+27+24+27= 186 coordinate points for the Quadrant I

186*4=744 coordinate points for all four Quadrants

y axis points: (0,1)..(0,16) =16 points (0,-1)...(0,-16)=16 points

x-axis points (1,0)...(16,0)=16 points (-1,0)...(-16,0)=16 points

Origin (0,0)= 1 coordinate point

Notice that I'm only counting the points on the circle not on the circumference, as the question asks.

Adding it all up:

744 +16+16+16+16+1=809 coordinate points

2) Quantity B: 77 coordinate points

Similarly:

x²+y²=5²

x²+y²=25

Points on the y-axis: 8 coordinate points. Points on x-axis: 8 coordinate points

(1,1)...(1,4) =4 points (2,1)...(2,4) =4 points (3,1)..(3,4)=4 points (4,1)...(4,3) =3 points

In Quadrant I = 15

In 4 quadrants = 60 coordinates points

60 coordinates points +4+4+4+4+1=77 coordinate points

Final comments:

We could solve it this only estimating. Just based on the radii of those circles. Since the question does not ask any quantity, but just the relationship. This could be a time saver way.