1.

The plane `x / 2 + y / 3 + z / 4 = 1` cuts the co-ordinate axes in `A, B, C`: then the area of the `DeltaABC` isA. `sqrt(29)` sq. unitsB. `sqrt(41)` sq. unitsC. `sqrt(61)` sq. unitsD. none of these

Answer» Correct Answer - C
The given plane cuts the coordinate axes in `A(2,0,0),B(0,3,0)` and `C(0,0,4)`.
`:.` Area of `DeltaABC=1/2ABxxACxxsin/_BAC`
Now,
`AB=sqrt(4+9+0)=sqrt(13),AC=sqrt(4+0+16)=sqrt(20)`
`cos /_BCA(vec(AB).vec(AC))/(|vec(AB)||vec(AC)|)=((-2hati+3hatj).(-2hati+4hatk))/(sqrt(4+9)sqrt(4+16))`
`impliescos /_BAC=(4+0+0)/(sqrt(13)sqrt(20))=4/(sqrt(13)sqrt(20))=2/(sqrt(65))`
`impliessin /_BAC=sqrt(1-4/65)=sqrt(61/65)`
Hence
Area of `DeltaABC=1/2xxsqrt(13)xxsqrt(20)xxsqrt(61/65)=sqrt(61)` sq. units.


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