Straight Line Question 72
Question: If straight lines $ ax+by+p=0 $ and $ x\cos \alpha +y\sin \alpha -p=0 $ include an angle $ \pi /4 $ between them and meet the straight line $ x\sin \alpha -y\cos \alpha =0 $ in the same point, then the value of $ a^{2}+b^{2} $ is equal to
Options:
A)1
B)2
C)3
D)4
Correct Answer: BShow Answer
Answer:
Solution:
Þ $ a\cos \alpha +b\sin \alpha =-a\sin \alpha +b\cos \alpha $ …..(i) It is given that the lines $ ax+by+p=0 $ , $ x\cos \alpha +y\sin \alpha -p=0 $ and $ x\sin \alpha -y\cos \alpha =0 $ are concurrent. \ $ | \begin{matrix} a & b & p \\ \cos \alpha & \sin \alpha & -p \\ \sin \alpha & -\cos \alpha & 0 \\ \end{matrix} |=0 $
Þ $ -ap\cos \alpha -bp\sin \alpha -p=0\Rightarrow -a\cos \alpha -b\sin \alpha =1 $
Þ $ a\cos \alpha +b\sin \alpha =-1 $ ……(ii)From (i) and (ii), $ -a\sin \alpha +b\cos \alpha =-1 $ From (ii) and (iii), $ {{(a\cos \alpha +b\sin \alpha )}^{2}}+{{(-a\sin \alpha +b\cos \alpha )}^{2}}=2 $
Þ $ a^{2}+b^{2}=2 $ .
BETA
