Question:
Find the coordinates of the point where the line joining the points $(2, -3, 1)$ and $(3, - 4, - 5)$ cuts the plane $2x + y + z = 7$.
Find the coordinates of the point where the line joining the points $(2, -3, 1)$ and $(3, - 4, - 5)$ cuts the plane $2x + y + z = 7$.
Updated On: Dec 1, 2024
- (1, 2, - 7)
- (1, - 2, 7)
- (-1, - 2, 7)
- (1, 2, 7)
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The Correct Option is B
Solution and Explanation
The direction ratios of the line are $3 - 2, - 4 - (-3), - 5 -1 $
i.e. $1, -1, - 6$
Hence equation of the line joining the given points.
is $\frac{x-2}{1} = \frac{y+3}{-1} = \frac{z-1}{-6} = r$ (say)
Coordinates of any point on this line are
$(r + 2, - r - 3, - 6r + 1)$
If this point lies on the given plane $2x + y + z = 7$,
then $2(r + 2) + (- r - 3) + (- 6r + 1) = 7 $
$\Rightarrow r = - 1$
Coordinates of any point on this line are
$(- 1 + 2, - (- 1) - 3, - 6 (-1) + 1)$ i.e. $(1, - 2, 7)$
i.e. $1, -1, - 6$
Hence equation of the line joining the given points.
is $\frac{x-2}{1} = \frac{y+3}{-1} = \frac{z-1}{-6} = r$ (say)
Coordinates of any point on this line are
$(r + 2, - r - 3, - 6r + 1)$
If this point lies on the given plane $2x + y + z = 7$,
then $2(r + 2) + (- r - 3) + (- 6r + 1) = 7 $
$\Rightarrow r = - 1$
Coordinates of any point on this line are
$(- 1 + 2, - (- 1) - 3, - 6 (-1) + 1)$ i.e. $(1, - 2, 7)$
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