Find the equations of the circles which touch the axis of x at a distance of 4 from the origin and cut off an intercept of 6 from the axis of y
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Answers
Let O BE THE CENTRE OF THE REQUIRED CIRCLE
SINCE IT TOUCHES THE CENTRE OF THE CIRCLE ,THE PERPENDICULAR FROM THE CENTRE,IS THE RADIUS OF THE CIRCLE
LET A AND B BE THE FOOT OF THE PERPENDICULARS FROM THE CENTRE TO THE COORDINATE AXES RESPECTIVELY
ALSO GIVEN THAT,THE CIRCLE CUTS OFF AN INTERCEPTS OF 6 FROM THE Y-AXIS
LET CD BE THE CHORD WHICH LIES ALONG Y-AXIS ,
CD=6 UNITS
THE LINE FROM THE CENTER BISECTS THE CHORD ,
SINCE OB BISECTS CD,WE HAVE ,BC=BD=CD/2
THEREFORE CB/2=6/2=3 UNITS
CONSIDER A RIGHT TRIANGLE,OCB
BY PYTHAGORAS THEORUM
WE HAVE,
OC^2=4^2+3^2
OC^2=16+9
OC^2=25
OC=5UNITS
THUS RADIUS OF THE CIRCLE IS 5UNITS
THE COORDINATES OF CENTRE OF CIRCLE IS O[4,5],0{DASH}[-4,5]
THUS THE EQUATION OF A CIRCLE
[X-4]^2+[Y-5]^2=5^2
X^2+16-8X+Y^2+25-10Y=25
X^2+16-8X+Y^2-10Y=O
OR X^2+Y^2+8X-10Y+16=0
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