find the locus of the midpoint of chord of circle x^2=y^2=9 such that the segment intercepted by the chord on the curve subtend right angle at the origin
Dear prasum,
I assume that you had a little trouble with your + = key
So lets say this is what you intended X^2 + Y^2= 9.The locus of the midpoint of chord which subtends a 90 deg arc is a circle of reduced diameter as I interpret a poorly worded question.
bjh
Hello, prasum!
Find the locus of the midpoint of chord of circle $\displaystyle x^2+y^2\:=\:9$
such that the segment intercepted by the chord subtends a right angle at the origin.
I'll try to explain this without a diagram.
The circle is centered at the origin $\displaystyle O$ and has radius 3.
Since the chord subtends $\displaystyle 90^o$ at the origin,
. . the chord subtends a quarter-circle.
Let one such chord connect $\displaystyle A(3,0)$ and $\displaystyle B(0,3)$.
. . Its midpoint is: .$\displaystyle M\left(\frac{3}{2},\:\frac{3}{2}\right)$
The distance $\displaystyle MO$ is: $\displaystyle \frac{3}{2}\sqrt{2}$
The locus is the circle centered at $\displaystyle O$ with radius $\displaystyle \frac{3}{2}\sqrt{2}$
. . Its equation is: .$\displaystyle x^2 + y^2 \:=\:\frac{9}{2}$
Hello Soroban,
All your answers are beautifly done including this one.My answer was enough so that the questioner could take it from there. And in looking at it thr apothem of the chord is simply 3/rad2
bjh