Question

From the point A (0, 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is :

Answer 1

@kreative_5

Answer 2

@goformit100

Answer 3

@RaphaelFilgueiras

Answer 4

@Girl_Ninja

Answer 5

@Nuggets2013

Answer 6

@RadEn

Answer 7

@dumbcow question: B is arbitrary point?

Answer 8

yeah seems like it can be any other point on the circle

Answer 9

and B is midpoint of AM?

Answer 10

yes
im thinking you can find the length of AB by using law of cosines
\[AB = \sqrt{2^{2} + 2^{2}-2(2)(2)\cos \theta}\]
since the radius of circle is 2
where -pi <theta < pi

Answer 11

I think when B move from A and goes around the circle , so M changing depends on the location of B . |AB| goes from 0 to 2, so, |AM| go from 0 to 4, hihi... think so only, not know how to put everything in logic

Answer 12

ok drawing looks about right, so can we assume locus is a circle of radius 4 ?
maybe

Answer 13

I think so, let O is the center of small circle, when B at A, the other end of diameter is A' at C( I call it is center of big circle) B moves, M move but 2 radius don't change

Answer 14

hi friend, I just try drawing out some locations of M as M1, M2, M3.... to observe what its orbit is, I see it as a circle like what I draw, you check, please

Answer 15

ok i just checked some points ... it works
also i should have noticed this before, proportionally it makes sense that if you double the chord length , you double the radius as well....essentially the triangle doubles in size

so equation of locus of M is circle:
\[(x+4)^{2} + (y-3)^{2} = 16\]

Answer 16

yes, that's what I think,

Answer 17

:)