The equation of the tangent to the circle x^2+y^2=16 such...

The equation of the tangent to the circle $x^{2}+y^{2}=16$ such that the corresponding normal passes through $(2,4)$ is
(a) $2 x-y \pm 4 \sqrt{5}=0$
(b) $x+2 y \pm 4 \sqrt{5}=0$
(c) $\quad x-2 y \pm 4 \sqrt{5}=0$
(d) $2 x+y \pm 4 \sqrt{5}=0$

Key Concepts

Normal to a Circle

The normal to a circle at a given point is the line that passes through that point and the center of the circle. Since it is perpendicular to the tangent at the point of contact, the normal’s direction is directly linked to the geometric positioning of the circle’s center and the point on the circle.

Condition for Tangency

For a line to be tangent to a circle, the perpendicular distance from the circle's center to the line must exactly equal the radius of the circle. This condition is used to derive relationships between the line's parameters and the circle's dimensions, ensuring that the line touches the circle at exactly one point.

Equation of a Circle

A circle in the Cartesian plane is defined by an equation of the form x² + y² = r², where (0,0) is the center and r is the radius. This standard form is foundational for understanding properties and relationships involving circles, including tangents and normals.

Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. At the point of tangency, the tangent is perpendicular to the radius drawn to that point. This property is crucial when determining the tangent's equation and establishing conditions such as tangency from external lines.

Collinearity and the Passage of a Line Through a Given Point

When an additional condition is imposed, such as a line (in this case, the normal) passing through a specific external point, it introduces a requirement of collinearity among the circle's center, the point of tangency, and the external point. This condition helps in determining the exact orientation and position of the tangent or normal by linking the geometry of the circle with external constraints.

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