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reply to the question shown in the image …...............................

reply to the question shown in the image …...............................


1 Answers

Aditya Gupta
2080 Points
2 years ago
hello abhishek. this is a really easy ques.
let the direction cosines be (l,m,n). or direction ratios are equivalently (1,m/l, n/l).
let m/l=k and n/l=p
so we are given a+bk+cp=0 and u+vk^2+wp^2=0
these are two equations in two variables k and p. eliminate k to obtain a quadratic eqn in just p. similarly eliminate p to obtain a quadratic eqn in just k.
now if the two lines are perpendicular then (1,k1,p1).(1,k2,p2)=0, where (.) is the dot product.
or 1+k1k2+p1p2=0
since we have a quad eqn in k, we can easily find the product of its roots k1k2 and similarly for p1p2.
plug in those values and there you have it!
for the parallel case, obviously (1,k1,p1)= λ(1,k2,p2)
1=λ, k1=λk2 and p1=λp2
or k1=k2 and p1=p2
now, for the roots of any quad eqn Ax^2+Bx+C=0 to be equal, we have B^2=4AC.
apply this above condition and you will obtain the necessary result!

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