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Mathematics Question on Fundamental Theorem of Calculus

Find the coordinates of the point where the line joining the points (2,−3,1)(2, -3, 1) and (3,−4,−5)(3, - 4, - 5) cuts the plane 2x+y+z=72x + y + z = 7.

A

(1, 2, - 7)

B

(1, - 2, 7)

C

(-1, - 2, 7)

D

(1, 2, 7)

Answer

(1, - 2, 7)

Explanation

Solution

The direction ratios of the line are 3−2,−4−(−3),−5−13 - 2, - 4 - (-3), - 5 -1
i.e. 1,−1,−61, -1, - 6
Hence equation of the line joining the given points.
is x−21=y+3−1=z−1−6=r\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)(r + 2, - r - 3, - 6r + 1)
If this point lies on the given plane 2x+y+z=72x + y + z = 7,
then 2(r+2)+(−r−3)+(−6r+1)=72(r + 2) + (- r - 3) + (- 6r + 1) = 7
⇒r=−1\Rightarrow r = - 1
Coordinates of any point on this line are
(−1+2,−(−1)−3,−6(−1)+1)(- 1 + 2, - (- 1) - 3, - 6 (-1) + 1) i.e. (1,−2,7)(1, - 2, 7)