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

 Consider the equations of two  intersecting straight lines   { ((ax+by+c=0)),((a_1 x+b_1 y+c_1 =0)) :}  Find the equation of straight line  passing through a given point  (x_0 ,y_0 ) and the intersection point  of the given straight lines.

$$\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{intersecting}\:\mathrm{straight}\:\mathrm{lines} \\ $$$$\begin{cases}{{ax}+{by}+{c}=\mathrm{0}}\\{{a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} =\mathrm{0}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{passing}\:\mathrm{through}\:\mathrm{a}\:\mathrm{given}\:\mathrm{point} \\ $$$$\left(\mathrm{x}_{\mathrm{0}} ,\mathrm{y}_{\mathrm{0}} \right)\:\mathrm{and}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{given}\:\mathrm{straight}\:\mathrm{lines}. \\ $$

Commented by mathocean1 last updated on 25/Feb/21

Hello  is it a_1 x+b_1 y+c_1 =0  the second eq?

$${Hello} \\ $$$${is}\:{it}\:{a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} =\mathrm{0}\:\:{the}\:{second}\:{eq}? \\ $$

Commented by bemath last updated on 25/Feb/21

yes

$$\mathrm{yes} \\ $$

Answered by EDWIN88 last updated on 25/Feb/21

The straight line specified by the equation  λ(ax+by+c)+μ(a_1 x+b_1 y+c_1 )=0  passes through the point of intersection  of the given straight lines. We required that  it should pass also through (x_0 ,y_0 ).  Then the equation of desired straight line  will 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

$$\mathrm{The}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{specified}\:\mathrm{by}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\lambda\left({ax}+{by}+{c}\right)+\mu\left({a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} \right)=\mathrm{0} \\ $$$${passes}\:{through}\:{the}\:{point}\:{of}\:{intersection} \\ $$$${of}\:{the}\:{given}\:{straight}\:{lines}.\:\mathrm{We}\:\mathrm{required}\:\mathrm{that} \\ $$$$\mathrm{it}\:\mathrm{should}\:\mathrm{pass}\:\mathrm{also}\:\mathrm{through}\:\left(\mathrm{x}_{\mathrm{0}} ,\mathrm{y}_{\mathrm{0}} \right). \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{desired}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{will}\:\mathrm{be}\:\left({ax}+{by}+{c}\right)\left({a}_{\mathrm{1}} {x}_{\mathrm{0}} +{b}_{\mathrm{1}} {y}_{\mathrm{0}} +{c}\right)−\left({a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} \right)\left({ax}_{\mathrm{0}} +{by}_{\mathrm{0}} +{c}\right)=\:\mathrm{0}\: \\ $$

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