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Question Number 133885 by bemath last updated on 25/Feb/21

$$\mathrm{Consider}\mathrm{the}\mathrm{equations}\mathrm{of}\mathrm{two}$$$$\mathrm{intersecting}\mathrm{straight}\mathrm{lines}$$$$\{\begin{array}{l}ax+by+c=0\\ a_{1}x+b_{1}y+c_1=0\end{array}$$$$\mathrm{Find}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{straight}\mathrm{line}$$$$\mathrm{passing}\mathrm{through}\mathrm{a}\mathrm{given}\mathrm{point}$$$$({\mathrm{x}}_0,{\mathrm{y}}_0)\mathrm{and}\mathrm{the}\mathrm{intersection}\mathrm{point}$$$$\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{straight}\mathrm{lines}.$$

Commented by mathocean1 last updated on 25/Feb/21

$$Hello$$$$isit{a}_{1}x+b_{1}y+c_1=0thesecondeq?$$

Commented by bemath last updated on 25/Feb/21

$$\mathrm{yes}$$

Answered by EDWIN88 last updated on 25/Feb/21

$$\mathrm{The}\mathrm{straight}\mathrm{line}\mathrm{specified}\mathrm{by}\mathrm{the}\mathrm{equation}$$$$\lambda (ax+by+c)+\mu (a_{1}x+b_{1}y+c_1)=0$$$$passesthroughthepointofintersection$$$$ofthegivenstraightlines.\mathrm{We}\mathrm{required}\mathrm{that}$$$$\mathrm{it}\mathrm{should}\mathrm{pass}\mathrm{also}\mathrm{through}({\mathrm{x}}_0,{\mathrm{y}}_0).$$$$\mathrm{Then}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{desired}\mathrm{straight}\mathrm{line}$$$$\mathrm{will}\mathrm{be}(ax+by+c)(a_1{x}_0+b_1{y}_0+c)-(a_{1}x+b_{1}y+c_1)({ax}_0+{by}_0+c)=0$$

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