99 Percentile Qs Bank for JEE MainMathematicsComplex Number
If (z=x+i y, x, y R ) and if the point (P ) in the argand plane represents (z ), then the locus of (P ) satisfying the condition ( ( z-1 z-3 i )= 2 ), is
Options
- A( z C / |z- 1+3 i 2 |= 10 2 )
- B( z C /(3-i) z+(3+i) z -6=0 )
- C( z C /(3-i) z+(3+i) z -6 > 0 ), ( . |z- 1+3 i 2 |= 10 2 )
- D( z C /(3-i) z+(3+i) z -6 < 0 ),
Correct answer
C. ( z C /(3-i) z+(3+i) z -6 > 0 ), ( . |z- 1+3 i 2 |= 10 2 )
Step-by-step solution
We have, ( ( z-1 z-3 i )= 2 ) ( array lc & (z-1)- (z-3 i)= 2 & [(x-1)+i y]- [x+(y-3) i]= 2 & ⁻¹ y x-1 - ⁻¹ y-3 x = 2 & ⁻¹ [ y 1+ y x-1 y-3 x ]= 2 & x y-(x-1)(x-3) x(x-1)+y(x-3) = 2 & x y-(x-1)(y-3) x(x-1)+y(y-3) = 1 0 & x(x-1)+y(y-3)=0 & x y=0 array ) ( aligned & (x- 1 2 )^2+ (y- 3 2 )^2= 1 4 + 9 4 & (x- 1 2 )^2+ (y- 3 2 )^2= ( 10 2 )^2 aligned ) Which is a circle with centre ( ( 1 2 , 3 2 ) ) and radius ( 10 2 ). ( z C:(3-i) z+(3+i) z -6 > 0, |z- 1+3 i 2 |= 10 2 )