If z=2+t+i √(3-t^2) where, t is real, the locus of the...

If $z=2+t+i \sqrt{3-t^{2}}$ where, $t$ is real, the locus of the points $z$ for different values of $t$ is the (a) circle with centre $(3,0)$ and radius $2 .$ (b) line segment joining the points $(0,1)$ and $(0,-1)$ (c) circle passing through the points $(1,0),(0,1)$ and $(0,-1)$ (d) circle of radius $\sqrt{3}$ passing through $(2-\sqrt{3}, 0)$ and $(2+\sqrt{3}, 0)$.

If $z=2+t+i \sqrt{3-t^{2}}$ where, $t$ is real, the locus of the points $z$ for different values of $t$ is the
(a) circle with centre $(3,0)$ and radius $2 .$
(b) line segment joining the points $(0,1)$ and $(0,-1)$
(c) circle passing through the points $(1,0),(0,1)$ and $(0,-1)$
(d) circle of radius $\sqrt{3}$ passing through $(2-\sqrt{3}, 0)$ and $(2+\sqrt{3}, 0)$.

Key Concepts

Elimination of the Parameter

Eliminating the parameter involves combining the parametric equations to remove the parameter, resulting in a Cartesian equation that relates the x and y coordinates directly. This process is useful for identifying the geometric nature of the locus, such as determining whether the points form a line, circle, or another conic section.

Circle Geometry in the Complex Plane

A circle in the complex plane is typically described by an equation of the form (x ? a)² + (y ? b)² = r², where (a, b) is the center and r is the radius. Recognizing when the Cartesian equation obtained after eliminating the parameter fits this form is key to understanding the overall geometry of the set of complex numbers described by a parametric equation.

Complex Numbers Representation

Complex numbers are expressed in the form x + iy, where x represents the real part and y represents the imaginary part. This form allows us to interpret algebraic expressions geometrically as points in a two-dimensional plane, with the x-axis representing the real component and the y-axis representing the imaginary component.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of one or more parameters. In the context of complex numbers, a parameter can be used to describe the real and imaginary parts independently, thus defining the locus of points that satisfy a particular condition as the parameter varies.

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