please help ive look at this question for hours and got no where. find the radii of the 2 circles which have their centre at the origin and which touch the circle x^2 + y^2 - 8x - 6y + 21 = 0 ??

thanks for the help

by completing the square I go

(x-4)^2 + (y-3)^2 = 4
The circle has centre (4,3) and radius 2
Sketch it
Now draw 2 circles with centre at the origin, one to touch the given circle on the outside and one to touch it on the inside.
The line from the origin through (4,3) will run through the points where the circles touch.
By Pythagoras the distance from (0,0) to (4,3) is 5
Since the radius of the given circle is 2,
the radius of the smaller circle at the origin has to be 3, and the radius of the larger circle has to be 7

thanks so much

To find the radii of the two circles that touch the given circle x^2 + y^2 - 8x - 6y + 21 = 0, we can follow these steps:

Step 1: Write the equation of the given circle in standard form.
Step 2: Find the distance between the center of the given circle and the origin.
Step 3: Find the difference between the radius of the given circle and the distance found in step 2. This will give you the radius of one of the circles.
Step 4: Find the sum of the radius of the given circle and the distance found in step 2. This will give you the radius of the other circle.

Let's go through each step in detail:

Step 1: Write the equation of the given circle in standard form.
The given equation is x^2 + y^2 - 8x - 6y + 21 = 0. We can rewrite it as:
(x^2 - 8x) + (y^2 - 6y) = -21
To complete the square, we need to add and subtract half of the coefficients of x and y terms separately:
(x^2 - 8x + 16) + (y^2 - 6y + 9) = -21 + 16 + 9
(x - 4)^2 + (y - 3)^2 = 4

Now, we have the equation of the given circle in standard form: (x - 4)^2 + (y - 3)^2 = 4

Step 2: Find the distance between the center of the given circle and the origin.
The center of the given circle is at coordinates (4, 3), which is the point (h, k) in the standard form of a circle equation. The distance between this center and the origin (0, 0) is given by the formula:
distance = √((h - 0)^2 + (k - 0)^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5

So, the distance between the center of the given circle and the origin is 5 units.

Step 3: Find the difference between the radius of the given circle and the distance found in step 2.
The radius of the given circle is 2 units (from the equation: (x - 4)^2 + (y - 3)^2 = 4). Now, subtract the distance found in step 2 from the radius to find the radius of one of the circles:
2 - 5 = -3

Therefore, the radius of one of the circles is -3 units.

Step 4: Find the sum of the radius of the given circle and the distance found in step 2.
The radius of the given circle is 2 units. Now, add the distance found in step 2 to the radius to find the radius of the other circle:
2 + 5 = 7

Therefore, the radius of the other circle is 7 units.

To summarize, the radii of the two circles, which have their center at the origin and touch the circle x^2 + y^2 - 8x - 6y + 21 = 0, are -3 units and 7 units.