A circle has a center of (1, 2) and passes through (1, –3). The circl

It's always a good idea to draw a diagram in co-ordinate geometry. The point (1 , -3) will be vertically below (1, 2) and hence you see that the radius will be (2 - (-3)) = 5

The circle will pass through every point which is 5 units away from (1, 2)

A. (–4, 2)
Distance from (1, 2) \(= \sqrt{(1 - (-4))^2 + (2 - 2)^2} = 5\)

B. (–3, 5)
Distance from (1, 2) \(= \sqrt{(1 - (-3))^2 + (2 - 5)^2} = 5\)

C. (0, 6)
Distance from (1, 2) \(= \sqrt{(1 - 0)^2 + (2 - 6)^2} = \sqrt{17}\)
Since, this point is not at a distance 5 away from (1, 2), it doesn't lie on the circle.

D. (4, –2)
E. (5, 5)

Answer (C)

we have (a -x)^2+ (b-y)^2=R^2 since the center (1;2) and passes thr (1;-3) then (1-1)^2+(2-(-3))^2=R^2 =>R=5
we have (1-x)^2+(2-y)^2=5^2
Using back solving we check for models 3 4 5; 4 3 5 ; 0 5 5; 5 0 5
then C is the answer

MAGOOSH OFFICIAL SOLUTION:

The radius is r = 5. First of all, 5 above the center, the circle goes through (1, 7) on the top, and 5 to the left & right of the center, the circle goes through (–4, 2) and (6, 2) on the same horizontal line as the center. That first point is choice (A).

That length of 5 can also be the hypotenuse of a 3-4-5 slope triangle, so starting from the center (1, 2), we could go over ±3 and up ±4, or over ±4 and up ±5. This means the circle must go through

right 3, up 4 = (4, 6)

right 4, up 3 = (5, 5) = option (E)

right 3, down 4 = (4, –2) = option (D)

right 4, down 3 = (5, –1)

left 3, up 4 = (–2, 6)

left 4, up 3 = (–3, 5) = option (D)

left 3, down 4 = (–2, –2)

left 4, down 3 = (–3, –1)

That’s all the points other than option (C). Notice that (–2, 6) and (4, 6) are on the circle, so another point between them, on the same horizontal line, (0, 6), could not be on the circle.

Answer = (C)
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Drawing figure can be really handy here. Once on paper, you can see radius is 5units.

Now, apply distance formula to calculate distance between two points keeping center of circle as one fixed point and points in options as variable point. On calculating you can see point C distance is ~ 4units. Which is inside the circle, hence it will not touch the circle.

Hope it was helpful !

Use the Rule => Radius Must be constant
so Using the distance formula => C
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The coordinates of any point through which a circle passes must satisfy the general equation of a circle:

\((x - h)^2 + (y - k)^2 = r^2\), where \(r\) is the circle's radius and \((h, k)\) are the coordinates of the circle's center.

In this case, the equation is \((x - 1)^2 + (y - 2)^2 = r^2\)

Because we are given one point, we can find the value of the squared radius easily by substituting the x and y coordinates of the given point (1, -3) into the equation:

\((1 - 1)^2 + (-3 - 2)^2 = r^2\)
\(0^2 + (-5)^2 = r^2\), or \(25 = r^2\) such that by substitution back into the general equation

\((x - 1)^2 + (y - 2)^2 = 25\)

Now just plug the coordinates from the answer choices in to see if they satisfy the equation.

(A) (-4, 2) yields \((-4 - 1)^2 + (2 - 2)^2\)=
25 + 0 = 25 KEEP

(B) (-3, 5) yields\((-3- 1)^2 + (5 - 2)^2\) =
16 + 9 = 25. KEEP

(C) (0, 6) yields \((0 - 1)^2 + (6 - 2)^2\)=
1 + 16 = 17. REJECT

(D) (4, -2) is the mirror image of A (and yields 9 + 16 = 25) KEEP

(E) (5, 5) yields (5 - 1)^2 + (5 - 2)^2 =
16 + 9 = 25. KEEP

Answer C

I'm not a fan of the distance formula.
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Given: A circle has a center of (1, 2) and passes through (1, –3).

Asked: The circle passes through all of the following EXCEPT:

Equation of the circle: -
(x-1)^2 + (y-2)^2 = (1-1)^2 + (2+3)^2 = 5^2 = 25

A. (–4, 2) : (-4-1)^2 + (2-2)^2 = 5^2: The circle passes through this point.
B. (–3, 5) : (-3-1)^2 + (5-2)^2 = 4^2 + 3^2 = 5^2: The circle passes through this point.
C. (0, 6) : (0-1)^2 + (6-2)^2 = 1^2 + 4^2 = 17: The circle DOES NOT pass through this point.
D. (4, –2) : (4-1)^2 + (-2-2)^2 = 3^2 + 4^2 = 5^2: The circle passes through this point.
E. (5, 5) : (5-1)^2 + (5-2)^2 = 4^2 + 3^2 = 5^2: The circle passes through this point.

IMO C

Like the majority of geometry questions, a little eyeballing and logic gets to the right answer really quickly without any "real" math. If we just plot the two points given, we can quickly see that the radius is five. Now plot a point five units up from the center, a point five units to the left, and a point five units to the right. Now try plotting the answer choices.

A. This is one of the points we already plotted. Wrong.
B. Hmmm, looks like it could work. If you spot that it makes a 3-4-5 triangle with the center of circle (it would make sense to look for this since concept among the answers since we know we have a radius of 5 and all of the answer choices have integer coordinates), great, but you don't even need to. If you're confident this point is on the circle, cross out B. If you're not, keep it.
C. Wait, this is one unit down and one unit over from the top of our circle. Surely that's not on the circle.

Answer choice C.