Let $P1:\overrightarrow{r}.(2\^i+\^j−3\^k )=4$ be a plane. Let $P_2$ be another plane which passes through 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 $16, α,β,$ then the value of $α + β$ is equal to _____ .

Question

Let  $P1:\overrightarrow{r}.(2\^i+\^j−3\^k )=4$  be a plane. Let  $P_2$  be another plane which passes through 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  $16, α,β,$  then the value of  $α + β$  is equal to _____ .

cdquestions Admin · Accepted Answer

Direction ratio of normal to $P_1≡< 2, 1, – 3 >$ and $P2≡\begin{vmatrix} \hat i & \hat j & \hat k \[0.3em] 0 & 1 & -5 \[0.3em] -1 & -2 & 5 \end{vmatrix}$ $P_2=−5\hat i−\hat j(−5)+\hat k(1)$ i.e. $< –5, 5, 1 >$ d.r’s of line of intersection are along vector $\begin{vmatrix} \hat i & \hat j & \hat k \[0.3em] 2 & 1 & -3 \[0.3em] -5 & 5 & 1 \end{vmatrix}$ $=\hat i(16)−\hat j(−13)+\hat k(15)$ i.e. $< 16, 13, 15 >$ Therefore, $α + β = 13 + 15 = 28$ So, the answer is $28$ .

Let $P1:\overrightarrow{r}.(2\^i+\^j−3\^k )=4$ be a plane. Let $P_2$ be another plane which passes through 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 $16, α,β,$ then the value of $α + β$ is equal to _____ .

Correct Answer: 28

Solution and Explanation

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Distance of a Point from a Plane