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
• B. True

Answer 1

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 $(x_1,y_1)$ and $(x_2,y_2)$ of its diameter are given, is given by,

$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$

Therefore equation of the circle whose diameter is AC is

$(x-3)(x-7)+(y-2)(y-10)=0$

$x^2-10x+21+y^2-12y+20=0$

$x^2+y^2-10x-12y+41=0$

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

L.H.S. = $x^2+y^2-10x-12y+41$

= $9^2+4^2-10\left(9\right)-12\left(4\right)+41$

= $81+16-90-48+41$

= $0$

= R.H.S.

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

Hence the given statement is false.