Pair Of Straight Lines Question 41
Question: If the ratio of gradients of the lines represented by $ ax^{2}+2hxy+by^{2}=0 $ is 1 : 3, then the value of the ratio $ h^{2}:ab $ is
[MP PET 1998]
Options:
A) $ \frac{1}{3} $
B) $ \frac{3}{4} $
C) $ \frac{4}{3} $
D) 1
Show Answer
Answer:
Correct Answer: C
Solution:
Gradients $ \frac{m_1}{m_2}=1:3 $
$ m_1=m,\ \ m_2=3m $
$ \therefore m_1+m_2=-\frac{2h}{b} $ …..(i) and $ m_1.m_2=\frac{a}{b} $ …(ii) From equation (i), $ m+3m=-\frac{2h}{b} $ or $ m=\frac{-h}{2b} $
From equation (ii), $ m.3m=\frac{a}{b} $
$ 3.\frac{h^{2}}{4b^{2}}=\frac{a}{b}\Rightarrow \frac{h^{2}}{ab}=\frac{4}{3} $ . Trick: If the gradients of two lines are in ratio $ 1:n $ .
Then $ \frac{h^{2}}{ab}=\frac{{{(n+1)}^{2}}}{4n}=\frac{{{(3+1)}^{2}}}{4.3}=\frac{4}{3} $ .
BETA
