The point `z` in the complex plane satisfying `|z+2|-|z-2|= ±3` lies on

Answer

`atoq,s;btor;c to p,q;d to q,r`.

Explanation

`atoq,s,btor,c to p,q,d to q,r`.
a. Tangent to the parabola having slope m is `ty=x+t^(2)`. It passes through the point (2,3). Then, `3t=2+t^(2)`, i.e., t=1 or 2.
The point of contact is (1,2) or (4,4)
b. Let a point on the circle be `P(x_(1),y_(1))`. Then the chord of contact of the parabola w.r.t P is `yy_(1)=2(x+x_(1))`. Comparing with y=2(x-2), we have `y_(1)=1andx_(1)=-2`, which also satisfy the circle.
c. Point Q on the parabola is at `(t^(2),2t)`.
Now, the area of triangle OPQ is
`|(1)/(2)|:(0,0),(4,-4),(t^(2),2t),(0,0):||=6or8t+4t^(2)=pm12`
For `t^(2)+2t-3=0,(t-1)(t+3)=0`. Then t=1 or t=-3.
Then point Q is (1,2) or (9,-6).
d. Point (1,2) and (-2,1) satisfy both the curves.