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
To find the radii of the two circles, we need to understand the geometric relationship between the given circle and the circles centered at the origin.
1. Start with the equation of the given circle: x^2 + y^2 - 8x - 6y + 21 = 0.
2. To make it easier to work with, let's complete the square for both x and y terms. Rearrange the equation by grouping the x and y terms:
(x^2 - 8x) + (y^2 - 6y) = -21
3. To complete the square for x, take half of the coefficient of x, square it, then add it to both sides of the equation:
(x^2 - 8x + 16) + (y^2 - 6y) = -21 + 16
(x - 4)^2 + (y^2 - 6y) = -5
4. Repeat the process for the y terms:
(x - 4)^2 + (y^2 - 6y + 9) = -5 + 9
(x - 4)^2 + (y - 3)^2 = 4
5. Now we have the equation of the given circle in the standard form:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) is the center of the circle and r is the radius.
6. Comparing the equation of the given circle to the standard form, we can see that the center is at (4, 3) and the radius is √4 or 2.
7. Since we need to find the radii of the two circles centered at the origin, we need to determine the distance between the origin (0, 0) and the center of the given circle (4, 3).
Using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(4 - 0)^2 + (3 - 0)^2]
= √[16 + 9]
= √25
= 5
8. The distance between the origin and the center of the given circle is 5.
Therefore, the radii of the two circles centered at the origin are 5 + 2 = 7 and 5 - 2 = 3, respectively.
Thus, the radii of the two circles are 7 and 3.