Question

In the figure below, 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. $2\frac{1}{14}cm$
• 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 lines (Contact of circles)
$YX\perp AB(Tangent\perp toradius)$
AX = 9, XB = 5 (Given)
$\Rightarrow AB=14,OB=OZ=7\left(sameradii\right);OX=7-5=2$
In triangle OXY, OY = 7 - r; YX = r, OX = 2
$\Rightarrow O{Y}^2=Y{X}^2+O{X}^2$ (Pythagoras' Theorem)
$(7-r{)}^2=r^2+2^2\Rightarrow 49-14r+r^2=r^2+4$
$\Rightarrow 14r=45\Rightarrow r=3\frac{3}{14}cm.$