Discussion
As shown in the figure, circle \(C_{1}\) with center \(O_{1}\) and radius \(r_{1}\) touches the square VWXY at points P and Q while circle \(C_{2}\) with center \(O_{2}\) and radius \(r_{2}\) touches the square VWXY at points R and S. The two circles touch each other at T.
Given \(r_{1} = 1\) cm and \(\overline{VY} = \overline{VW} = 4\) cm, \(r_{2}\) = _____ cm.
Answer: C
The radius is \(r_{2} = 7 - 4\sqrt{2}\) cm, about 1.34 cm.
Put the corner W at the origin with WX along the x-axis and WV along the y-axis, so the square runs from 0 to 4 in each direction. Circle \(C_{1}\) touches VW and WX, so its centre is \(O_{1} = (1, 1)\). Circle \(C_{2}\) touches VY and XY, the two sides meeting at Y, so \(O_{2} = (4 - r_{2},\ 4 - r_{2})\).
Both centres lie on the diagonal WY, and since the circles touch externally, \(O_{1}O_{2} = 1 + r_{2}\):
\[ \begin{aligned} \sqrt{2}\,(3 - r_{2}) &= 1 + r_{2} \\ r_{2}(1 + \sqrt{2}) &= 3\sqrt{2} - 1 \\ r_{2} &= (3\sqrt{2} - 1)(\sqrt{2} - 1) \\ &= 6 - 3\sqrt{2} - \sqrt{2} + 1 = 7 - 4\sqrt{2} \end{aligned} \]
The other values are impossible: \(4 - 3\sqrt{2}\) is negative, and \(1 + 2\sqrt{2} \approx 3.83\) and \(5 + 3\sqrt{2} \approx 9.24\) give circles wider than the 4 cm square.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development