Show that, x=1/2(3+5cosA), y=1/2(-4+5sinA) represent a circle passing through the origin.find co-ordinates of centre and length of radius of the circle?

#x=1/2(3+5cosA), y=1/2(-4+5sinA)#

1 Answer
Shwetank Mauria
Feb 17, 2018

Center is #(1.5,-1)# and radius is #2.5#

Explanation:

As #x=1/2(3+5cosA)#, we have

#cosA=(2x-3)/5# ............(1)

and as #y=1/2(-4+5sinA)#

#sinA=(2y+4)/5# ............(2)

Squaring and adding (1) and (2), we get

#(2x-3)^2/25+(2y+4)^2/25=1#

or #(2x-3)^2+(2y+4)^2=25# ............(3)

i.e. #4x^2-12x+9+4y^2+16y+16=25#

i.e. #4x^2+4y^2-12x+16y=0#

As the equation does not have any constant term, the origin i.e. point #(0,0)# satisfies the equation and hence circle passes through origin. Further, we can write (3) (by dividing each term by #4#) as

#(x-3/2)^2+(y+1)^2=(5/2)^2#

Hence the equation represents a circle with center #(3/2,-1)# i.e. #(1.5,-1)# and radius #5/2=2.5#

graph{4x^2+4y^2-12x+16y=0 [-4.352, 5.65, -4.24, 0.76]}

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