Given planes P1: cy+bz=xP2 : az+cx =yP3: bx+ay=z P1, P2 and P3 pass through one line, if(A) a2+b2+c2=ab+bc+ca(B) a2+b2+c2+2abc=1(C)a2+b2+c2=1(D) a2+b2+c2+2ab+2bc+2ca+2abc=1

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Grade 11Vectors

Given planes
P1: cy+bz=x
P2 : az+cx =y
P3: bx+ay=z

P1, P2 and P3 pass through one line, if
(A) a²+b²+c²=ab+bc+ca
(B) a²+b²+c²+2abc=1
(C)a²+b²+c²=1
(D) a²+b²+c²+2ab+2bc+2ca+2abc=1

Satviki Pathak

11 Years agoGrade 11

2 Answers

Yash Jain

11 Years ago

The answer to your question is B) a²+b²+c²+2abc=1.

I’ve got it using a bit long method so try to solve it with any better and easier one. I used the concept of family of Planes.What I did is as follows:

The equation of family of planes from P1 and P2 is given as,
-→ -x+cy+bz + L(cx-y+az) = 0
-→ (-1+Lc)x + (c-L)y + (b+La)z = 0 ---(i)

Since P3 also passes through the line of intersection, i.e.P1, P2 and P3 pass through one line,Thus the coeff. of P3 and that of eq (i) are proportional. So,
(-1+Lc)/b = (c-L)/a = (b+La)/-1 = k (say) ---(ii)
Now, lets start eliminating the assumed constants.
From eq (ii) we get L = (bk+1)/c {= c-ka = -(1+b)/a}--(iii)
From (iii), conclude that k = -(bc+a)/(ab+c)
Putting this value of k in the equation (bk+1)/c = c-ka, we’ll get to our answer which is...
a²+b²+c²+2abc=1
I am sure you’ll be able to find more convenient method for the same. Best wishes.