The length of the tangent segment drawn from the point...

The length of the tangent segment drawn from the point represented by $-1+\mathrm{i}$ to the circle $|\mathrm{z}-(3+4 \mathrm{i})|=2$ is
(a) $\sqrt{21}$
(b) $5+\sqrt{2}$
(c) $7-\sqrt{2}$
(d) $\sqrt{24}$

Key Concepts

Circle in the Complex Plane

A circle in the complex plane is defined by an equation of the form |z - c| = r, where c is the center of the circle and r is its radius. This representation translates the algebraic expression into a geometric locus of points that are all equidistant from the center, forming a circle.

Tangent to a Circle

A tangent to a circle is a line or segment that touches the circle at exactly one point. In the context of the complex plane, when a point outside a circle is connected to the circle by a tangent, the length of this tangent segment can be determined using the Pythagorean theorem, relating the distance from the point to the circle's center and the radius of the circle.

Complex Numbers

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane, where each complex number has a real and an imaginary part. This concept is fundamental in algebra and geometry, enabling the representation and analysis of points in a plane using a single algebraic expression.

Distance Formula in the Complex Plane

The distance between two complex numbers, analogous to points in the plane, is calculated using the Euclidean distance formula. This involves computing the modulus of the difference between the two complex numbers, which captures the geometric distance between their corresponding points.

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