SOLUTION: What is the smallest distance between the point(–2, –2) and a point on the circumference of the circle given by (x – 1)^2 + (y –2)^2 = 4? A) 3 B) 4 C) 5 D) 6

Algebra ->  Test -> SOLUTION: What is the smallest distance between the point(–2, –2) and a point on the circumference of the circle given by (x – 1)^2 + (y –2)^2 = 4? A) 3 B) 4 C) 5 D) 6      Log On


   

Question 1041650: What is the smallest distance between the point(–2, –2) and a point on the circumference of the circle given by (x – 1)^2 + (y –2)^2 = 4?
A) 3
B) 4
C) 5
D) 6
Found 3 solutions by josgarithmetic, robertb, MathTherapy:
Answer by josgarithmetic(39620) About Me  (Show Source):
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Think through the situation logically. Know the Distance Formula!

The center for your given circle is (1,2). The point outside(?) the circle is given as (-2,-2). Radius size for your circle is 2.

You are looking for two distances. One of them you already know and other of them you can calculate.

The point on the circle which is smallest distance from given point to the circle IS ON THE LINE which contains (-2,-2) and center(1,2). You are not asked to find this point, but you are asked for the size of the difference of distances.

Think carefully about all of that....


You are looking for THIS value:
----------------------------------
Distance from (-2,-2) to (1,2)   |
Minus                            |
4.                               |
----------------------------------

That's it!

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The line that passes through the point (-2,-2) and the center (1,2) has equation
y-2+=+%28%282--2%29%2F%281--2%29%29%28x-1%29 <==> y+=+%284%2F3%29x+%2B+2%2F3,
after simplifying.
This line will intersect the circle in two (diametrically) opposite points on the circle, one of which is the farthest from the point (-2,-2), and the other the nearest to the point (-2,-2).
To find out the points, substitute y+=+%284%2F3%29x+%2B+2%2F3 into
%28x-1%29%5E2+%2B+%28y-2%29%5E2+=+4.
===>
==> %28x-1%29%5E2+%2B++%2816%2F9%29%28x-1%29%5E2+=+4
==> %2825%2F9%29%28x-1%29%5E2+=+4 ==> %28x-1%29%5E2+=+36%2F25
==> x = 11/5 or -1/5.
Incidentally, these are the x-coordinates of the two diametrically opposite points on the circle that are farthest and nearest the point (-2,-2), respectively. The corresponding y-value is y = 2/5. The distance of this point (-1/5,2/5) from (-2,-2) is sqrt%28%28-1%2F5--2%29%5E2%2B%282%2F5--2%29%5E2%29+=+highlight%283%29.

Answer by MathTherapy(10553) About Me  (Show Source):
You can put this solution on YOUR website!
What is the smallest distance between the point(–2, –2) and a point on the circumference of the circle given by (x – 1)^2 + (y –2)^2 = 4?
A) 3
B) 4
C) 5
D) 6
Length of line, or distance from (- 2, - 2) to center of circle: (1, 2) = 5 units

Length of radius = sqrt%284%29, or 2 units

Shortest distance from (- 2, - 2) and circumference of circle: 5 - 2, or highlight_green%28matrix%281%2C4%2C+3%2C+units%2C+%22%28CHOICE%22%2C+%22A%29%22%29%29