Find the cartesian equation of a line passing through two points (8,-5,7) and (7,1,4).

(a) \(\frac{x-8}{1}=\frac{y+5}{-6}=\frac{z-7}{3}\)

(b) \(\frac{2x-8}{-1}=\frac{3y+5}{6}=\frac{4z-7}{-3}\)

(c) \(\frac{x-8}{-1}=\frac{y+5}{6}=\frac{z-7}{-3}\)

(d) \(\frac{x-8}{-2}=\frac{y+5}{5}=\frac{z-7}{-7}\)

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The question is from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

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Right answer is (c) \(\frac{x-8}{-1}=\frac{y+5}{6}=\frac{z-7}{-3}\)

Easy explanation: Consider A(8,-5,7) and B(7,1,4)

i.e. \((x_1,y_1,z_1)\)=(8,-5,7) and \((x_2,y_2,z_3)\)=(7,1,4)

The cartesian equation for a line passing through two points is given by

\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)

∴ the cartesian equation for the given line is \(\frac{x-8}{-1}=\frac{y+5}{6}=\frac{z-7}{-3}\)