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Step-by-Step Solution
Step 1: Identify the two given points
We are given two points in 3D space:
(5, 1, a) and (3, b, 1).
We want the equation of the line passing through these points.
Step 2: Write the parametric form of the line
The direction ratios of the line are obtained by subtracting coordinates of the two points:
Direction ratios = (3 − 5, b − 1, 1 − a) = (−2, b − 1, 1 − a).
Let the parameter of the line be $ \lambda $. Then the parametric equations for any point $ (x, y, z) $ on this line can be written as:
$ x - 5 = -2 \lambda, \quad y - 1 = (b - 1)\lambda, \quad z - a = (1 - a)\lambda. $
Thus, a general point $ P(\lambda) $ on the line is:
$ \bigl(-2\lambda + 5,\;(b - 1)\lambda + 1,\;(1 - a)\lambda + a\bigr). $
Step 3: Impose the condition for crossing the yz-plane
The yz-plane is given by $ x = 0 $. So we set
$ -2\lambda + 5 = 0 $
to find the parameter $ \lambda $ at the intersection with the yz-plane.
Solving for $ \lambda $ gives:
$ -2\lambda + 5 = 0 \quad \Rightarrow \quad \lambda = \frac{5}{2}. $
Step 4: Find the coordinates at the intersection with the yz-plane
Substitute $ \lambda = \frac{5}{2} $ into the parametric form of $ y $ and $ z $:
$ y = (b - 1)\left(\frac{5}{2}\right) + 1 $
$ z = (1 - a)\left(\frac{5}{2}\right) + a $
The intersection point with the yz-plane is also given to be
$ \left(0,\;\frac{17}{2},\;-\frac{13}{2}\right). $
Therefore,
$ (b - 1)\left(\frac{5}{2}\right) + 1 = \frac{17}{2}, \quad (1 - a)\left(\frac{5}{2}\right) + a = -\frac{13}{2}. $
Step 5: Solve the equations to find a and b
From
$ (b - 1)\left(\frac{5}{2}\right) + 1 = \frac{17}{2}, $
we have:
$ \frac{5}{2}(b - 1) + 1 = \frac{17}{2}
\quad \Longrightarrow \quad \frac{5}{2} (b - 1) = \frac{17}{2} - 1
\quad \Longrightarrow \quad \frac{5}{2} (b - 1) = \frac{15}{2}
\quad \Longrightarrow \quad b - 1 = 3
\quad \Longrightarrow \quad b = 4.
$
From
$ (1 - a)\left(\frac{5}{2}\right) + a = -\frac{13}{2}, $
we have:
$ \frac{5}{2} (1 - a) + a = -\frac{13}{2}
\quad \Longrightarrow \quad \frac{5}{2} - \frac{5a}{2} + a = -\frac{13}{2}
\quad \Longrightarrow \quad \frac{5}{2} + \left(a - \frac{5a}{2}\right) = -\frac{13}{2}
\quad \Longrightarrow \quad \frac{5}{2} - \frac{3a}{2} = -\frac{13}{2}.
$
Rearranging further:
$ -\frac{3a}{2} = -\frac{13}{2} - \frac{5}{2}
\quad = -\frac{18}{2}
\quad = -9
\quad \Longrightarrow \quad \frac{3a}{2} = 9
\quad \Longrightarrow \quad a = 6.
$
Step 6: Conclude the values of a and b
The values that satisfy the given condition are
$ a = 6 $
and
$ b = 4, $
which matches the correct answer among the options provided.
