Find the coordinate of the point P where the line through A(3,−4,−5) and B(2,−3,1) crosses the plane passing through three points L(2,2,1),M(3,0,1) and N(4,−1,0). Also, find the ratio in which P divides the line segment AB.
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Solution
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The equation of line AB is given by →r=(3^i−4^j−5^k)+λ(^i−^j−6^k)
The normal to the plane will be →LM×→MN=(^i−2^j)×(−^i+^j+^k)=0+^k−^j−2^k−2^i=−2^i−^j−^k
The equation of the plane is given by 2x+y+z+d=0
Since (2,2,1) lies on the plane, we get 4+2+1+d=0
⇒d=−7
The plane becomes 2x+y+z−7=0
Let the point P on the line be (3+t,−4−t,−5−6t)
Since this lies on the plane, 6+2t−4−t−5−6t−7=0
⇒−5t−10=0 or t=−2
P=(1,−2,7)
1=2m+3nm+n
⇒m+n=2m+3n or m=−2n
m:n=−2:1
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