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Question

(a) Check whether the circle with centre at point (2,4) and radius 5 units passes through the point (2,0).
(b) Write the co-ordinates of the points at which this circle cuts the x-axis.

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Solution

The general equation of a circle with centre (h,k) and radius r units is (xh)2+(yk)2=r2
Given: Centre of circle is (2,4) and radius =5 units
Thus, the equation of a given circle is
(x2)2+(y4)2=52
x24x+4+y28y+16=25
x2+y24x8y+20=25
x2+y24x8y5=0....(i)
Now, to check whether the point (2,0) passes through the circle or not, substitute x=2 and y=0 in the L.H.S. of equation (i).
Thus, we have L.H.S. =22+024×28×05
=4+0805
=90
Hence, the point (2,0) does not lie on the circle.
(b) The circle cuts the x-axis at y=0
Substituting the value y=0 in equation (i), we get
x2+024x8(0)5=0
x24x5=0
x25x+x5=0
x(x5)+1(x5)=0
(x5)(x+1)=0
x=1 or x=5
Thus, the co-ordinates of the points at which the circle cuts the x-axis are (1,0) and (5,0).

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