Let the lines
$${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$$
$${L_2}:\overrightarrow r = \left( {\widehat i + 3\widehat j + \widehat k} \right) + \mu \left( {\widehat i + \widehat j + 5\widehat k} \right);\,\mu \in R$$,
intersect at the point S. If a plane ax + by $-$ z + d = 0 passes through S and is parallel to both the lines L1 and L2, then the value of a + b + d is equal to ____________.
Answer (integer)
5
Solution
<p>As plane is parallel to both the lines we have d.r's of normal to the plane as <7, $-$2, $-$1></p>
<p>$$\left( {from\,\left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
1 & 2 & 3 \cr
1 & 1 & 5 \cr
} } \right| = 7\widehat i - \widehat j(2) + \widehat k( - 1)} \right)$$</p>
<p>Also point of intersection of lines is $2\widehat i + 4\widehat j + 6\widehat k$</p>
<p>$\therefore$ Equation of plane is</p>
<p>$7(x - 2) - 2(y - 4) - 1(z - 6) = 0$</p>
<p>$\Rightarrow 7x - 2y - z = 0$</p>
<p>$a + b + d = 7 - 2 + 0 = 5$</p>
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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