Question

A plane passing through the point (3,1,1) contains two lines whose direction ratios are (1, -2, 2) and (2, 3, -1) respectively. If this plane also passes through the point (α,-3,5), then α is equal to
A. -5
B. 10
C. 5
D. -10

Answer 1

Hint: The equation of the plane is passing through a point (m1, m2, m3). It is perpendicular to the vector with direction ratios p, q, r is given by, $(x - {m_1}).p + (y - {m_2}).q + (z - {m_3}).r = 0$

Complete step by step solution: 
Calculate the normal vector $\overline m $ of the plane. As a result, take into consideration the two direction ratios represented by the vectors $\widehat i - 2\widehat j + 2\widehat k$and $2\widehat i + 3\widehat j - \widehat k$
Calculate the cross product of the vectors $\widehat i - 2\widehat j + 2\widehat k$ and $2\widehat i + 3\widehat j - \widehat k$ then,
$\overline m $= ($\widehat i - 2\widehat j + 2\widehat k$) × ($2\widehat i + 3\widehat j - \widehat k$)
now we will form the matrix of the above equation
$\overline m $= \[\left| {\begin{array}{*{20}{c}}
  {\widehat i}&{\widehat j}&{\widehat k} \\
  1&{ - 2}&2 \\
  2&3&{ - 1}
\end{array}} \right|\]
Solving the matrix and finding the vector
=\[\widehat i({\mathbf{2}} - {\mathbf{6}}) - \widehat j( - {\mathbf{1}} - {\mathbf{4}}) + \widehat k({\mathbf{3}} + {\mathbf{4}})\]
$ = - 4\widehat i + 5\widehat j + 7\widehat k$
To obtain the equation of the needed plane, use the formula for the equation of a plane when a point passes through it and a vector is perpendicular to it.
$({m_1},{m_2},{m_3}) = (3,1,1)$, and $p = - 4,q = 5.r = 7$
Substitute these values in equation,
$(x - {m_1}).p + (y - {m_2}).q + (z - {m_3}).r = {0_{}}$
$(x - 3)( - 4) + (y - 1).5 + (z - 1).7 = 0$
$ - 4x + 12 + 5y - 5 + 7z - 7 = 0$
$ - 4x + 5y + 7z = 0$
Points $(a, - 3,5)$satisfy the equation. So,
\[ - 4x + 5y + 7z = 0\]
We will substitute the value of points in the equations
\[ - 4(a) + 5( - 3) + 7(5) = 0\]
Now we will solve the bracket
\[ - 4a - 15 + 35 = 0\]
\[4a = 20\]
\[a = \dfrac{{20}}{4}\]
Hence we will find the value of a
\[a = 5\]
Therefore, option (C) is correct.

Note: The majority of students make errors in determining the determinant, which causes the entire answer to be incorrect. Due to a lack of conceptual understanding, some students choose to immediately answer this question.