Triangles And Properties Of Triangle Question 39
Question: Given, that a, b, c are the sides of a triangle ABC which is right angles at, C then the minimum value of $ {{( \frac{c}{a}+\frac{c}{b} )}^{2}} $ is
Options:
A) 0
B) 4
C) 6
D) 8
Show Answer
Answer:
Correct Answer: D
Solution:
[d] $ a=c\sin \theta ,b=ccos\theta $
$ \Rightarrow {{( \frac{c}{a}+\frac{c}{b} )}^{2}} $ $ ={{( \frac{1}{\sin \theta }+\frac{1}{\cos \theta } )}^{2}}=\frac{4(1+sin2\theta )}{{{\sin }^{2}}2\theta } $ $ =4( \frac{1}{{{\sin }^{2}}2\theta }+\frac{1}{\sin 2\theta } ), $ Where $ 0<\theta <\frac{\pi }{2} $
$ \Rightarrow ,{{( \frac{c}{a}+\frac{c}{b} )}^{2}}_{\min }=8, $ when $ 2\theta =90{}^\circ . $
BETA
