Submitted by Elicias on Tue, 09/26/2017 - 13:10
Forums:
Hello,
I am currently working on the assignment 2 warm down, and I have found two expressions for $(OX)^2$ in $R^2$ and $r^2$ but after rearranging them, I keep ending up with $R^2 - r^2 = 2ab-b^2$. I know that I have clearly done something wrong algebraically or else I have misinterpreted the lengths of the sides involved.
I would appreciate any pointers on where my mistake could be.
Thank you for your help!
p.s. I hope the maths formatting works!
The maths formatting looks
The maths formatting looks great!
Assuming that $X$ is the point where the radius meets the line AB we have:
\[
(OX)^2 = (OB)^2 - (XB)^2 \quad \text{and} \quad (OX)^2=(OP)^2 - (XP)^2
\]
From the information in the question we have $OB=R$, $XB=a$, $OP=r$, $XP=b$. Using this
and equating the expressions for $(OX)^2$ gives us:
\[
R^2-a^2=r^2-b^2\\
R^2-r^2=a^2-b^2
\]
The area between the two circles is $\pi (R^2-r^2)=\pi(a^2-b^2)$
I think you had $PB=b$ rather
I think you had $PB=b$ rather than $XP=b$?
Aha!
Yes, that would make a lot of sense, it seem I had misread the question.
Thank you for your help!