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  • Can someone help me with this, - Complex numbers and quadratic equations - JEE Main-7

 Let \small z\epsilon C, the set of complex numbers.  Then the equation, \small 2\left |Z+3i \right |-\left | Z-i \right |=0 represents :

 

  • Option 1)

    a circle with radius \frac{8}{3}

  • Option 2)

    a circle with diameter \frac{10}{3}

  • Option 3)

     an ellipse with length of major axis \frac{16}{3}

     

  • Option 4)

    an ellipse with length of minor axis

    \frac{16}{9}

 

Answers (2)

best_answer

Using definition of Complex Numbers, Definition of Complex Number -

z=x+iy, x,y\epsilon R  & i2=-1

-

Real part of z = Re (z) = x & Imaginary part of z = Im (z) = y

 and,

Definition of Modulus of z(Complex Number) -

\left | z \right |=\sqrt{a^{2}+b^{2}} is the distance of z from origin in Argand plane

-

Real part of z = Re (z) = a & Imaginary part of z = Im (z) = b

 we have, 2|z+3i|-|z-i|=0\Rightarrow 2\sqrt{x^{2}+(y+3)^2}=\sqrt{x^{2}+(y-1)^2}

\Rightarrow 3x^2+3y^2+26y+35=0\Rightarrow x^2+y^2+(26/3)y+(35/3)=0

Therefore, radius of this circle is \sqrt {(13/3)^2-(35/3)}=8/3


Option 1)

a circle with radius \frac{8}{3}

This is the correct answer.

Option 2)

a circle with diameter \frac{10}{3}

Not a correct answer.

Option 3)

 an ellipse with length of major axis \frac{16}{3}

 

The equation is not that of an ellipse but a circle.

Option 4)

an ellipse with length of minor axis

\frac{16}{9}

The equation is not that of an ellipse but a circle.

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