Question

9) The distance of point (x₁,y₁) fro each of two straight lines which passes through the origin of co-ordinates is $\delta$; find the combined equations of these straight lines.

Answer 1

dear student

$$\begin{gathered} letthelinesthroughoriginbe \\ y-m_{1}x=0 \\ y-m_{2}x=0 \\ combinedequationis \\ {y}^2-(m_1+m_2)xy+m_1{m}_2{x}^2=0 \\ dis\tan ceofpoint(x_1,y_1)fromliney=mxis \\ dis\tan ce=\frac{\left|m{x}_1-y_1\right|}{\sqrt{1+m^2}} \\ \delta =\frac{\left|m{x}_1-y_1\right|}{\sqrt{1+m^2}} \\ \delta \sqrt{1+m^2}=\left|m{x}_1-y_1\right| \\ {\delta }^2(1+m^2)=m^2{x_1}^2+{y_1}^2-2m{x}_1{y}_1 \\ {m}^2({x_1}^2-{\delta }^2)-2m{x}_1{y}_1+{y_1}^2-{\delta }^2=0 \\ ifrootsare{m}_{1}and{m}_2 \\ {m}_1+m_2=\frac{2x_1{y}_1}{({x_1}^2-{\delta }^2)} \\ {m}_1*m_2=\frac{{y_1}^2-{\delta }^2}{({x_1}^2-{\delta }^2)} \\ {y}^2-(m_1+m_2)xy+m_1{m}_2{x}^2=0 \\ {y}^2-\frac{2x_1{y}_1}{({x_1}^2-{\delta }^2)}xy+\frac{{y_1}^2-{\delta }^2}{({x_1}^2-{\delta }^2)}{x}^2=0 \end{gathered}$$

regards