Question

Assertion :

Let $z_1,z_2,z_3$ satisfy $\left|\frac{z+2}{z-1}\right|=2$ and $z_0=2$. Consider least positive arguments wherever required

$2\arg \left(\frac{z_1-z_3}{z_2-z_3}\right)=\arg \left(\frac{z_1-z_0}{z_2-z_0}\right)$. Reason: $z_1,z_2,z_3$ satisfy $|z-z_0|=2$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$z_1,z_2,z_3$ satisfy $|z+2|=2|z-1|$
If $z$ is written as $x+iy,\sqrt{x^2+4x+4+y^2}=2\sqrt{x^2-2x+1+y^2}$
Squaring, we get $x^2+4x+4+y^2=4x^2-8x+4+4y^2$
$\Rightarrow 3x^2-12x+3y^2=0$ or $x^2-4x+y^2=0$
$\Rightarrow (x-2{)}^2+(y{)}^2=4$ is the required circle with centre as $(2,0)$ and radius $2.$
The assertion says that twice the angle subtended by a chord at a point on the circle equals the angle subtended by the same chord on the centre, which is true.
The reason says $|z-2|=2$ since $z_o=2$ implying option $A$ to be correct.