Question

Which of the following parameters given in the figure are enough to construct the figure (choose the minimum number of parameters possible):

Distance AO = d

Radius of the bigger circle (with center A) = R

Radius of the smaller circle (with center B) = r

$\angle$POA = $\theta$

Note that other than the parameters given in an option, nothing else is given and you have to come up with the figure as given above.

• A. R, r, d, AO, BO
• B. AO, BO, d
• C. R, d, r
• D. R, d, $\theta$

Answer 1

The correct option is C

R, d, r

Once R and d are given, we can construct the bigger circle (the one with centre A) and the tangents to the circle. So we are left with the inner circle.

Suppose we are given with the inner circle radius, r. Consider $△$PAO and $△$QBO. Seeing that AP $\parallel$ BQ, we can say that $△$PAO and $△$QBO are similar.

So, $\frac{R}{r}$ = $\frac{PA}{QB}$ = $\frac{AO}{BO}$ = $\frac{AB+BO}{BO}$ = $\frac{AB}{BO}$ + 1

$\Rightarrow$ $\frac{AB}{BO}$ = $\frac{R-r}{r}$

Now, we need to find the point which divides the line segment AO in the ratio (R-r): r. Once we do this, we get the point B. Now with B as center and radius r, draw a circle. Our construction is complete.

Hence we would require only 3 parameters, i.e. R, r and d.