MHT CET · Maths · Pair of Lines
If the lines \(x^2-4 x y+y^2=0\) make angles \(\alpha\) and \(\beta\) with positive direction of X-axis, then \(\cot ^2 \alpha+\cot ^2 \beta=\)
- A 14
- B 16
- C 18
- D 20
Answer & Solution
Correct Answer
(A) 14
Step-by-step Solution
Detailed explanation
We have lines \(x^2-4 x y+y^2=0\) and slopes of the lines are \(\tan \alpha\) and \(\tan \beta\).
\( \therefore \tan \alpha+\tan \beta=4 \text { and } \tan \alpha \cdot \tan \beta=1 \)
\( \therefore \tan \alpha=\frac{1}{\tan \beta}=\cot \beta \)
\( \therefore \cot \beta+\tan \beta=4 \text { squaring we get } \cot ^2 \beta+\tan ^2 \beta\)\(+2=16 \)
\( \Rightarrow \tan ^2 \beta+\cot ^2 \beta=14\)
\( \therefore \tan \alpha+\tan \beta=4 \text { and } \tan \alpha \cdot \tan \beta=1 \)
\( \therefore \tan \alpha=\frac{1}{\tan \beta}=\cot \beta \)
\( \therefore \cot \beta+\tan \beta=4 \text { squaring we get } \cot ^2 \beta+\tan ^2 \beta\)\(+2=16 \)
\( \Rightarrow \tan ^2 \beta+\cot ^2 \beta=14\)
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