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  • Please help! - Complex numbers and quadratic equations - JEE Main-9

The locus of the centre of a circle which touches the circle \left | z-z_{1} \right |= a\: and \: \left | z-z_{2} \right |= b externally

\dpi{100} \left ( z,z_{1},z_{2}\: are \: complex\: number\: \right ) will be

  • Option 1)

    an ellipse

  • Option 2)

    a hyperbola

  • Option 3)

    a circle

  • Option 4)

    none of these

 

Answers (2)

best_answer

As we have learned

Equation of circle -

\left |z-z_{0} \right |=r 

z_{0} = centre of circle

r= radius of circle

z lies on circle.

- wherein

Locus of z will be a circle as z is always at a fixed distance r from a fixed point z_{0}

z=x+iy,  z_{0}=x_{0}+iy_{0}

 

 

 

\left | z-z_1 \right |= a+r..............(1)

\left | z-z_2 \right |= b+r..............(2)

On (1)-(2) \rightarrow  

\left | z-z_1 \right |-\left | z-z_2 \right |= a-b

which is the locus of a hyperbola

 

 

 

 

 


Option 1)

an ellipse

Option 2)

a hyperbola

Option 3)

a circle

Option 4)

none of these

Posted by

Himanshu

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