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एक रेखा, एक घन के विकर्णों के साथ alph...

एक रेखा, एक घन के विकर्णों के साथ `alpha, beta, gamma, delta`, कोण बनती है तो सिद्ध कीजिए कि
` " " cos ^(2) alpha + cos ^(2) beta + cos ^(2) gamma + cos ^(2) delta = ( 4)/(3)`

A

1

B

`4//3`

C

`3//4`

D

`4//5`

Text Solution

Verified by Experts

The correct Answer is:
B
Let the DC's of given line be l,m,n. Diagonals of cube are OP,AL,BM,CN. The DC's of the diagonals are `(1/sqrt3,1/sqrt3,1/sqrt3),((-1)/sqrt3,1/sqrt3,1/sqrt3),(1/sqrt3,1/sqrt3,1/sqrt3),(1/sqrt3,(-1)/sqrt3,1/sqrt3),(1/sqrt3,1/sqrt3,(-1)/sqrt3)`
Let the line makes the angles with the diagonals OA,AL,BM and CN are respectively, `alpha,beta,gamma and delta`
`therefore" "cosalpha=(l+m+n)/sqrt3,cosbeta=(-l+m+n)/sqrt3`
`cosgamma=(l-m+n)/sqrt3,cosgamma=(l+m-n)/sqrt3`
`thereforecos^2alpha+cos^2beta+cos^2gamma+cos^2delta=4/3(l^2+m^2+n^2)=4/3`
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