Let the plane ax + by + cz = d pass through (2, 3, $-$5) and is perpendicular to the planes
2x + y $-$ 5z = 10 and 3x + 5y $-$ 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to :
Solution
<p>Equation of pane through point (2, 3, $-$5) and perpendicular to planes 2x + y $-$ 5z = 10 and 3x + 5y $-$ 7z = 12 is</p>
<p>$$\left| {\matrix{
{x - 2} & {y - 3} & {z + 5} \cr
2 & 1 & { - 5} \cr
3 & 5 & { - 7} \cr
} } \right| = 0$$</p>
<p>$\therefore$ Equation of plane is $(x - 2)( - 7 + 25) - (y - 3)$</p>
<p>$( - 14 + 15) + (z + 5)\,.\,7 = 0$</p>
<p>$\therefore$ $18x - y + 7z + 2 = 0$</p>
<p>$\Rightarrow 18x - y + 7z = - 2$</p>
<p>$\therefore$ $- 18x + y - 7z = 2$</p>
<p>On comparing with $ax + by + cz = d$ where d > 0 is a = $-$ 18, b = 1, c = $-$ 7, d = 2</p>
<p>$\therefore$ $a + 7b + c + 20d = 22$</p>
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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