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 (x−h)2+(y−k)2=r2Given: Centre of circle is (2,4) and radius =5 unitsThus, the equation of a given circle is(x−2)2+(y−4)2=52⇒x2−4x+4+y2−8y+16=25⇒x2+y2−4x−8y+20=25⇒x2+y2−4x−8y−5=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+02−4×2−8×0−5=4+0−8−0−5=−9≠0Hence, the point (2,0) does not lie on the circle.(b) The circle cuts the x-axis at y=0Substituting the value y=0 in equation (i), we getx2+02−4x−8(0)−5=0⇒x2−4x−5=0⇒x2−5x+x−5=0⇒x(x−5)+1(x−5)=0⇒(x−5)(x+1)=0⇒x=−1 or x=5Thus, the co-ordinates of the points at which the circle cuts the x-axis are (−1,0) and (5,0).

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