If the equation of the plane passing through the line of intersection of the planes 2x $-$ 7y + 4z $-$ 3 = 0, 3x $-$ 5y + 4z + 11 = 0 and the point ($-$2, 1, 3) is ax + by + cz $-$ 7 = 0, then the value of 2a + b + c $-$ 7 is ____________.
Answer (integer)
4
Solution
Equation of plane can be written using family of planes : P<sub>1</sub> + $\lambda$P<sub>2</sub> = 0<br><br>(2x $-$ 7y + 4z $-$ 3) + $\lambda$ (3x $-$ 5y + 4z + 11) = 0<br><br>It passes through ($-$2, 1, 3)<br><br>$\therefore$ ($-$4 + 7 + 12 $-$ 3) + $\lambda$ ($-$6 $-$ 5 + 12 + 11) = 0<br><br>$-$2 + $\lambda$ (12) = 0<br><br>$\lambda$ = ${1 \over 6}$.<br><br>$\therefore$ 12x $-$ 42y + 24z $-$ 18 + 3x $-$ 5y + 4z + 11 = 0<br><br>15x $-$ 47y + 28z $-$ 7 = 0<br><br>$\therefore$ a = 15, b = $-$47, c = 28<br><br>$\therefore$ 2a + b + c $-$ 7 = 30 $-$ 47 + 28 $-$ 7 = 4
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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