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  • The condition that one of the straight lines given by the equation <img alt="\mathrm{ax}^2+2 \mathrm{hxy}+\mathrm{by}^2=0" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bax%7D%5E2

The condition that one of the straight lines given by the equation \mathrm{ax}^2+2 \mathrm{hxy}+\mathrm{by}^2=0 may coincide with one of those given by the equation \mathrm{a^{\prime} x^2+2 h^{\prime} x y+b^{\prime} y^2=0 } \mathrm{is \left(a b^{\prime}-a^{\prime} b\right)^2= \mathrm{k}\left(\mathrm{ha}^{\prime}-\mathrm{ah} \mathrm{h}^{\prime}\right)\left(\mathrm{bh}^{\prime}-\mathrm{b}^{\prime} \mathrm{h}\right)}, where \mathrm{k}=

Option: 1

4


Option: 2

3


Option: 3

2


Option: 4

1


Answers (1)

best_answer

Sol: Let an equation of the common line be

\mathrm{y=m_1 x . \Rightarrow b m l^2+2 h m_1+a=0 and b^{\prime} m^2+ 2 h^{\prime} m_1+a^{\prime}=0}

\mathrm{\Rightarrow \frac{m_1^2}{2\left(h a^{\prime}-a h^{\prime}\right)}=\frac{m_1}{a b^{\prime}-a^{\prime} b}=\frac{1}{2\left(b h^{\prime}-b^{\prime} h\right)}}

\mathrm{\Rightarrow \frac{h^{\prime}-a h^{\prime}}{b h^{\prime}-b^{\prime} h}=\left[\frac{a b^{\prime}-a^{\prime} b}{2\left(b^{\prime}-b^{\prime} h\right)}\right]^2}

\mathrm{ \Rightarrow\left(a b^{\prime}-a^{\prime} b\right)^2=4\left(h a^{\prime}-a h^{\prime}\right)\left(b h^{\prime}-b^{\prime} h\right) }
which is the required condition.

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chirag

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