A plane P-ax-by-cz-1 passes through the intersection of planes r- - k-3 and r-k-2- If plane P divides the line segment joining M3- 0- 2 and N0- 3- -1 in the ratio 2 - 1 internally- then a - b - c is equal to

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Family of Planes Passing through the Intersection of Two Planes

A plane P:ax+...

Question

A plane $P : a x + b y + c z = 1$ passes through the intersection of planes $\to r \cdot (^i +^j +^k) = - 3$ and $\to r \cdot (^i -^j +^k) = 2.$ If plane $P$ divides the line segment joining $M (3, 0, 2)$ and $N (0, 3, - 1)$ in the ratio $2 : 1$ internally, then $(a + b + c)$ is equal to

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Solution

$P_{1} : x + y + z + 3 = 0$
$P_{2} : x - y + z - 2 = 0$
Equation of plane $P$ passing through the intersection of planes $P_{1}$ and $P_{2}$ is
$P_{1} + λ P_{2} = 0$
$\Rightarrow (x + y + z + 3) + λ (x - y + z - 2) = 0 \dots (1)$

Let point $R$ divide the line segment $M N$ in $2 : 1$ internally, then
$R \equiv (\frac{2 (0) + 1 (3)}{2 + 1}, \frac{2 (3) + 1 (0)}{2 + 1}, \frac{2 (- 1) + 1 (2)}{2 + 1})$
$\Rightarrow R \equiv (1, 2, 0)$

As point $R$ lies on the plane $P,$
$\Rightarrow 6 + λ (- 3) = 0$
$\Rightarrow λ = 2$

Hence, equation of plane $P$ is $3 x - y + 3 z = 1$
$∴ a + b + c = 3 - 1 + 3 = 5$

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A plane P:ax+by+cz=1 passes through the intersection of planes →r⋅(^i+^j+^k)=−3 and →r⋅(^i−^j+^k)=2. If plane P divides the line segment joining M(3,0,2) and N(0,3,−1) in the ratio 2:1 internally, then (a+b+c) is equal to

A plane $P : a x + b y + c z = 1$ passes through the intersection of planes $\to r \cdot (^i +^j +^k) = - 3$ and $\to r \cdot (^i -^j +^k) = 2.$ If plane $P$ divides the line segment joining $M (3, 0, 2)$ and $N (0, 3, - 1)$ in the ratio $2 : 1$ internally, then $(a + b + c)$ is equal to