Let P_1:-> r . ( 2hat i + hat j - 3hat k ) = 4 be a plane. Let P_(2) be another plane which passes through the points (2, -3, 2), (2, -2, -3) and (1, -4, 2). If the direction ratios of the line of intersection of P_(1) and P_(2) be 16, alpha, beta, then the value of alpha + beta is equal to ________________.

Direction ratio of normal to P_1 equiv and that of P_2 equiv hat i hat j hat k \ 0 1 -5 \ -1 -2 5 = -5hat i - hat j(-5) + hat k(1) i.e. d.r's of line of intersection are along vector hat i hat j hat k \ 2 1 -3 \ -5 5 1 = hat i(16) - hat j(-13) + hat k(15) i.e. Therefore alpha + beta = 13 + 15 = 28

Let P1 r 2 i j 3 k 4 be a plane Let P2 be another plane... | null (Mathematics)

Direction ratio of normal to

{P_1} \equiv < 2,1, - 3 >

and that of

P_2 \equiv \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & -5 \\ -1 & -2 & 5 \end{vmatrix} = -5\hat{i} - \hat{j}(-5) + \hat{k}(1)

\text{i.e.} < - 5,5,1 >

d.r's of line of intersection are along vector

\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -3 \\ -5 & 5 & 1 \end{vmatrix} = \hat{i}(16) - \hat{j}(-13) + \hat{k}(15)

\text{i.e.} < 16,13,15 >

\therefore \alpha + \beta = 13 + 15 = 28

Explanation

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