The line passing through the points (a, 1, 6) | vedclass

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Question

(A) The equation of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{

Vedclass · Accepted Answer

$19$

Vedclass · Answer

(A) The equation of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$.Substituting the points $(a, 1, 6)$ and $(3, 4, b)$,we get $\frac{x - a}{3 - a} = \frac{y - 1}{4 - 1} = \frac{z - 6}{b - 6}$.This line passes through the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$.Substituting $x = 0$ into the equation: $\frac{0 - a}{3 - a} = \frac{y - 1}{3} = \frac{z - 6}{b - 6}$.Using the $y$-coordinate: $\frac{\frac{17}{2} - 1}{3} = \frac{\frac{15}{2}}{3} = \frac{5}{2}$.Now,equate the ratios: $\frac{-a}{3 - a} = \frac{5}{2} \implies -2a = 15 - 5a \implies 3a = 15 \implies a = 5$.Next,use the $z$-coordinate: $\frac{\frac{-13}{2} - 6}{b - 6} = \frac{5}{2} \implies \frac{\frac{-25}{2}}{b - 6} = \frac{5}{2} \implies \frac{-25}{b - 6} = 5 \implies -5 = b - 6 \implies b = 1$.Finally,calculate $(3a + 4b) = 3(5) + 4(1) = 15 + 4 = 19$.

The line passing through the points $(a, 1, 6)$ and $(3, 4, b)$ crosses the $yz$-plane at $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$. Then the value of $(3a + 4b)$ is:

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