Find coordinates of the point, which when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ forms a line that is parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ ?

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Find coordinates of the point, which when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ forms a line that is parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ ?

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Answer 1 · 2016-09-26T16:55:59

Any point lying on $(b_{2} - b_{1}) (x - c_{1}) - (y - c_{2}) (a_{2} - a_{1}) = 0$ $(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0$ when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ would form a line parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ .

Explanation:

The slope of a line passing through $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ is $\frac{b_{2} - b_{1}}{a_{2} - a_{1}}$ $(b_2-b_1)/(a_2-a_1)$

The equation of a line with a slope $m$ $m$ and passing through $(x_{1}, y_{1})$ $(x_1,y_1)$ is $y - y_{1} = m (x - x_{1})$ $y-y_1=m(x-x_1)$

As the slope of the line parallel to above too would be $\frac{b_{2} - b_{1}}{a_{2} - a_{1}}$ $(b_2-b_1)/(a_2-a_1)$ and as it passes through $(c_{1}, c_{2})$ $(c_1,c_2)$ , its equation would be

$y - c_{2} = \frac{b_{2} - b_{1}}{a_{2} - a_{1}} (x - c_{1})$ $y-c_2=(b_2-b_1)/(a_2-a_1)(x-c_1)$ or

$(b_{2} - b_{1}) (x - c_{1}) - (y - c_{2}) (a_{2} - a_{1}) = 0$ $(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0$

Hence any point lying on $(b_{2} - b_{1}) (x - c_{1}) - (y - c_{2}) (a_{2} - a_{1}) = 0$ $(b_2-b_1)(x-c_1)-(y-c_2)(a_2-a_1)=0$ will satisfy the given condition, i.e. when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ would form a line parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ .

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Find coordinates of the point, which when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ forms a line that is parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ ?

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Find coordinates of the point, which when joined with (c1,c2)(c_1,c_2) forms a line that is parallel to the line joining (a1,a2)(a_1,a_2) and (b1,b2)(b_1,b_2)?

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Explanation:

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Find coordinates of the point, which when joined with $(c_{1}, c_{2})$ $(c_1,c_2)$ forms a line that is parallel to the line joining $(a_{1}, a_{2})$ $(a_1,a_2)$ and $(b_{1}, b_{2})$ $(b_1,b_2)$ ?