Question

In the adjoining figure, X is a point on diameter AB of the circle with centre O such that AX = 9 cm, XB = 5 cm. Find the radius of the circle (centre Y) which touches the diameter at X and touches the circle, centre O, internally at Z.

• A. $3\frac{3}{14}cm$
• B. $3\frac{1}{14}cm$
• C. $1\frac{1}{14}cm$
• D. $2\frac{3}{14}cm$

Answer 1

The correct option is A $3\frac{3}{14}cm$

Let YX = YZ = r (same radii); OYZ is a straight line (contact of circles)
$YX\perp AB$ (Tangent $\perp$ to radius);
AX = 9, XB = 5 (given)
$\Rightarrow AB=14,OB=OZ=7$ (Same radii)
$OX=7-5=2$
In triangle $OXY,OY=7-r,YX=r,OX=2$
$\Rightarrow O{Y}^2=Y{X}^2+O{X}^2$ (Pythagoras' Theorem)
$(7-r{)}^2=r^2+2^2=49-14r+r^2=r^2+445=14r=3\frac{3}{14}cm$.