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If the component lines whose combined equation is ax^2-bxy-y^2=0 makes angle \phi\,\,\, and \ \theta with x-axis, then find the value of  \tan (\phi + \theta)

Option: 1

\frac{ab}{1+a}\\


Option: 2

\frac{-ab}{1+a}\\


Option: 3

\frac{ab}{1-a}\\


Option: 4

None of these


Answers (1)

best_answer

\\\text{Given Equation is }ax^2-bxy-y^2=0 \\\\ \text{Let }m_1(=\tan \phi), m_2(=\tan \theta) \text{ be the slopes of the lines} \\ \tan(\phi+\theta)=\frac{m_1+m_2}{1-m_1m_2} \\\\ \text{As we know, for } a x^{2}+2 h x y+b y^{2}=0\\ \begin{array}{l}{\mathrm{m}_{1}+\mathrm{m}_{2}=-\frac{2 \mathrm{h}}{\mathrm{b}}} \\ {\mathrm{m}_{1} \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{b}}}\end{array}\\So,\,\, \mathrm{m}_{1}+\mathrm{m}_{2}=-\frac{-b}{-1}=-b\\ {\mathrm{m}_{1} \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{-1}}}\\ \tan(\phi+\theta)=\frac{-b}{1+a}

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Pankaj

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