Question

The equation of a circle is (x + 12)2 + (y + 16)2 = (r1)2, and the circle passes through the origin. The equation of the circle then changes to (x – 30)2 + (y – 16)2 = (r2)2, and the circle still passes through the origin. What are the values of r1 and r2?

Answer 1

A)
r1 = 10 and r2 = 17
Eliminate


B)
r1 = 10 and r2 = 34



C)
r1 = 20 and r2 = 17



D)
r1 = 20 and r2 = 34

Answer 2

sorry, the "eliminate" was not supposed to be there

Answer 3

so... what's the center for the 1st equation of the circle?

Answer 4

(-12, -16)?

Answer 5

ok... so...what's the center for the 2nd equation of the circle?

Answer 6

is it (30, 16)?

Answer 7

what next?

Answer 8

right
they both pass through the origin, so we want to know their radius

so you have 2 points for each radius, the center of the circle, and the origin, to find the radius, or distance between then \(\bf \textit{distance between 2 points}\\ \quad \\
\begin{array}{lllll}
&x_1&y_1&x_2&y_2\\
&(-12\quad ,&-16)\quad &(0\quad ,&0)\\ \quad \\
&(30\quad ,&16)\quad &(0\quad ,&0)
\end{array}\qquad
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
\)

Answer 9

An alternative solution:
Both circles pass through the origin.
Therefore, x = 0, y = 0 must satisfy the equations.
Put (0,0) in the first equation and find r1
Put (0,0) in the second equation and find r2

Answer 10

yes.

Answer 11

Thank you both :) one more thing: how would you solve this?

Find the equation of the parabola with focus (-4, 0) and directrix x = 4.

Answer 12

sorry bout the "eliminate" thing again

Answer 13

One definition of a parabola is: The distance of any point on the parabola to the focus = distance of the point from the directrix:

Let (x,y) represent a generic point on the parabola.
The distance of the point (x, y) from (-4,0) = ?

Distance of the point (x,y) from the line x = 4 is what?

Equate them both to find the equation of the parabola.

Answer 14

so would it be x^2=16y?

Answer 15

No. Try again.

Answer 16

yes.

Answer 17

thank you :-)

Answer 18

you are welcome. :)