Find the equation of plane passing through the point $(1, 2, 3)$ and through the intersecting line of the planes $\to r . (^i + 2^j + 3^k) 5 = 0$ and $\to r . (2^i - 4^j +^k) = 3$

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Solution

The equation of the plane which passes through the line of intersection of given plane is
$\to r = (^i + 2^j + 3^k) 5 + l [\to r (2^i - 4^j +^k) - 3] = 0..... (1)$
It passes through the point $(1, 2, 3)$ or $(1^i + 2^j + 3^k)$
$(^i + 2^j + 3^k) (^i + 2^j + 3^k) + 5 + l (^i + 2^j + 3^k) (2^i - 4^j +^k) = 0$
$1 + 4 + 9 + 5 + l [(2 - 68 - 3 + 3)] = 0$
$19 + l (- 12) = 0$
$l = \frac{19}{12}$
Putting value of $l$ in equation (1)
$\to r (^i + 2^j + 3^k) + 5 + \frac{19}{12} \to r (2^i - 4^j +^k) - 3 = 0$
$\to r (12^i + 24^j + 3^k + 38^i - 76^j + 19^k) + 2$
$\to r (50^i + 52^j + 22^k) + 2 = 0$

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Similar questions

Q. Prove that equation of the plane passing through (1, 2, 3) and perpendicular to planes

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and

\to r . (2^i + 3^j + 4^k) = 0

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y

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Find the equation of the plane through the line of intersection of the planes x + y +z =1 and 2x + 3y + 4z =5 which is perpendicular to the plane x - y + z = 0. Also find the distance of the plane obtained above, from the origin.

Find the distance of the point (2, 12, 5) from the point of intersection of the line $\to r = 2^i - 4^j + 2^k + λ (3^i + 4^j + 2^k)$ and the plane $\to r . (^i - 2^j +^k) = 0$ .

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Find the equation of plane passing through the point (1,2,3) and through the intersecting line of the planes →r.(^i+2^j+3^k)5=0 and →r.(2^i−4^j+^k)=3

Find the equation of plane passing through the point $(1, 2, 3)$ and through the intersecting line of the planes $\to r . (^i + 2^j + 3^k) 5 = 0$ and $\to r . (2^i - 4^j +^k) = 3$