यदि एक सरल रेखा एक घन के विकर्णो के साथ `alpha,beta,gamma` और `delta` के कोण बनती है तो `cos^(2)alpha+cos^(2)beta+cos^(2)gamma+cos^(2)delta` बराबर है

Options

A. `1//2`
B. `1//4`
C. `3//4`
D. `4//3`

Answer

D

Explanation

माना विकर्ण `OP=a(hati+hatj+hatk)`
`AL=a(hatj+hatk-hati)`
`CF=a(hati+hatj-hatk)`
माना `OL=xhati+yhatj+zhatk`

`Rightarrow cos alpha=(x+y+z)/(sqrt(3(x^(2)+y^(2)+z^(2))))`
`cos beta=(x+y-z)/(sqrt(3(x^(2)+y^(2)+z^(2))))`
`cos gamma=(y+z-x)/(sqrt(3(x^(2)+y^(2)+z^(2))))`
`cos delta=(z+x-y)/(sqrt(3(x^(2)+y^(2)+z^(2))))`
`therefore cos^(2)alpha+cos^(2)beta+cos^(2)gamma+cos^(2)delta`
`=([{:(,(x+y+z)^(2)+(x+y-z)^(2)),(,+(y+z-x)^(2)+(z+x-y)^(2)):}])/(3(x^(2)+y^(2)+z^(2)))`
`=(4(x^(2)+y^(2)+z^(2)))/(3(x^(2)+y^(2)+z^(2)))=(4)/(3)`