Question

A circle is drawn with AB as the diameter whose endpoints are $A(-4,2)$ and $B(8,10)$. If another circle with diameter one-third of the above circle is drawn with the same centre, what are the points that the circle cuts AB?

• A. $(6,\frac{20}{3})$ and $(-2,\frac{11}{3})$
• B. $(6,\frac{22}{3})$ and $(-2,\frac{14}{3})$
• C. $(4,\frac{22}{3})$ and $(0,\frac{14}{3})$
• D. $(4,\frac{20}{3})$ and $(1,\frac{14}{3})$

Answer 1

The correct option is C

$(4,\frac{22}{3})$ and $(0,\frac{14}{3})$

Given, diameter AB has the endpoints as $(-4,2)$ and $(8,10)$.

We shall first find the centre of this circle.

We know that the centre of the circle '$O(x,y)$' is the mid-point of AB.

Hence by mid-point formula, we have,

$(x,y)$ = $(\frac{-4+8}{2},\frac{2+10}{2})$ = $(2,6)$

Now, if another circle with diameter one-third of the above circle is drawn with the same centre, then OP:PB = 1:2

Then by section formula, we get,

$x$-coordinate of P = $2+\frac{1}{3}(8-2)$ = $4$

$y$-coordinate of P = $6+\frac{1}{3}(10-6)$ = $\frac{22}{3}$

Therefore coordinates of P are $(4,\frac{22}{3})$

Since coordinates of P are $(4,\frac{22}{3})$ and $(2,6)$ is the centre of the circle, we must have coordinates of point Q as $(0,\frac{14}{3})$.