Pair Of Straight Lines Question 126
Question: The equation $ x^{2}-3xy+\lambda y^{2}+3x-5y+2=0 $ when $ \lambda $ is a real number, represents a pair of straight lines. If $ \theta $ is the angle between the lines, then $ cose{c^{2}}\theta $ =
[EAMCET 1992]
Options:
3
9
10
100
Show Answer
Answer:
Correct Answer: C
Solution:
The equation $ x^{2}-3xy+\lambda y^{2}+3x-5y+2=0 $ represents a pair of straight lines.
$ \therefore 2\lambda +2( -\frac{5}{2} )( \frac{3}{2} )( -\frac{3}{2} )-\frac{25}{4}-\frac{9\lambda }{4}-\frac{18}{4}=0 $
$ \Rightarrow \lambda =2 $
If $ \theta $ is the angle between the lines, then $ \tan \theta =\frac{2\sqrt{h^{2}-ab}}{a+b}=\frac{2\sqrt{(9/4)-2}}{1+2}=\frac{2\sqrt{1/4}}{3}=\frac{1}{3} $
$ \Rightarrow \csc^{2}\theta =1+\cot^{2}\theta =1+9=10 $
BETA
