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Question

Consider the points A(3,2),B(9,4),C(7,10). Then the circle with AC as diameter doesn't pass through B.


A

False

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B

True

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Solution

The correct option is A

False


In order to find whether the circle with AC as diameter passes through B or not, we need to first find the equation of the circle whose diameter is AC and plug in the coordinates of point B in this equation. If the coordinates satisfy the equation, then B is a point on the circle. Otherwise, B is not a point on the circle.

We know that the equation of a circle when the end points (x1,y1) and (x2,y2) of its diameter are given, is given by,

(xx1)(xx2)+(yy1)(yy2)=0

Therefore equation of the circle whose diameter is AC is

(x3)(x7)+(y2)(y10)=0

x210x+21+y212y+20=0

x2+y210x12y+41=0

Now we shall check if B(9,4) lies on the circle.

L.H.S. = x2+y210x12y+41

= 92+4210(9)12(4)+41

= 81+169048+41

= 0

= R.H.S.

Therefore (9,4) is a point on this circle.

Hence the given statement is false.


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