JEE MAIN - Mathematics (2023 - 25th January Morning Shift - No. 12)
Let $$\mathrm{z_1=2+3i}$$ and $$\mathrm{z_2=3+4i}$$. The set $$\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}$$ represents a
hyperbola with the length of the transverse axis 7
hyperbola with eccentricity 2
straight line with the sum of its intercepts on the coordinate axes equals $$-18$$
straight line with the sum of its intercepts on the coordinate axes equals $$14$$
Explanation
$\left|z-z_{1}\right|^{2}-\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2}$
$\Rightarrow(x-2)^{2}+(y-3)^{2}-(x-3)^{2}-(y-4)^{2}=1+1$
$\Rightarrow-4 x+4+9-6 y-9+6 x-16+8 y=2$
$\Rightarrow 2 x+2 y=14$
$\Rightarrow x+y=7$
$\Rightarrow(x-2)^{2}+(y-3)^{2}-(x-3)^{2}-(y-4)^{2}=1+1$
$\Rightarrow-4 x+4+9-6 y-9+6 x-16+8 y=2$
$\Rightarrow 2 x+2 y=14$
$\Rightarrow x+y=7$
Comments (0)
