Let a plane `a x+b y+c z+1=0,w h e r ea ,b ,c`are parameters, make anangle `60^0`with the line `x=y=z ,45^0`with the line `x=y-z=0`and`theta`with the plane `x=0.`The distance of the plane from point `(2,1,1)`is 3 units. Findthe value of `theta`and the equation ofthe plane.

Let a plane `a x+b y+c z+1=0,w h e r ea ,b ,c`are parameters, make ana

1.	Let a plane `a x+b y+c z+1=0,w h e r ea ,b ,c`are parameters, make anangle `60^0`with the line `x=y=z ,45^0`with the line `x=y-z=0`and`theta`with the plane `x=0.`The distance of the plane from point `(2,1,1)`is 3 units. Findthe value of `theta`and the equation ofthe plane.
Answer» Correct Answer - `theta= 60^(@) , 2x+ (2pm sqrt2) y + (2 pm sqrt2) z + 4 =0, 2x+ (2pm sqrt2)y + (2 pm sqrt2)z - 20 =0` `ax+by+cz+1=0" "(i)` It makes an `angle60^(@)` with the line x=y=z. So we get `sin60^(@)=(a+b+c)/(sqrt3suma^(2))` or `3sqrtsuma^(2)=2(a+b+c)" "(ii)` Plane (i) makes an angle of `45^(@)` with the line `x=y-z=0(or(x)/(0)=(y)/(1)=(z)/(1))` `sin45^(@)=(b+c)/(sqrt2sqrt(suma^(2)))orsqrt(suma^(2))=a+c" "(iii)` Plane (i) makes an `angletheta` with plane x=0. So we get `costheta=(a)/(sqrt(suma^(2)))" "(iv)` From (ii) and (iii) we get `(sqrtsuma^(2))=2a` or `(a)/(sqrtsuma^(2))=(1)/(2)` From (iv), `costheta=1//2ortheta=60^(@)` Distance of plane (i) from the point (2,1,1) is 3 units. `implies(2a+b+c+1)/(sqrt(suma^(2)))=+-3` or `+-3sqrt(suma^(2))=2a+b+c+1` Case I: `+-3sqrt(suma^(2))=2a+b+c+1" "(v)` From (ii) and (iv), we get `b+c-1=0" "(vi)` and from (iii) and (iv), we get `2a+b+c+1=3(b+c)" "(vii)` From (vi) and (vii), we get `a=(1)/(2),b =(2pmsqrt2)/(4)andc=(2pmsqrt2)/(4)` Hence, the set of such planes is Case II: `-3sqrtsuma^(2)=2a+b+c+1` `a=(-1)/(10),b=(-(2pmsqrt2))/(20) andc=(-(2pmsqrt2))/(20)` Hence, the other set of the planes is `2x+(2pmsqrt2)y+(2pmsqrt2)z-20=0.`