Two points are (𝑥1 , 𝑦1) and (𝑥2 , 𝑦2).
Slope =m=x2−x1y2−y1
Intercept=c=x2−x1x1y2−x2y1
Now, equation is y=mx+c
⇒y=x2−x1y2−y1x+x2−x1x1y2−x2y1 ...(i)
Now, given determinant is ∣∣xx1x2yy1y2111∣∣=0
On expanding,
x(y1−y2)−y(x1−x2)+(x1y2−x2y1)=0⇒x(x1−x2)(y1−y2)−y+(x1−x2)(x1y2−x2y1)=0⇒y=x2−x1y2−y1x+x2−x1x1y2−x2y1 ...(ii)
From (i) & (ii), it is proved.
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