The coordinates of a point on the line y=2 from which the...

The coordinates of a point on the line $y=2$ from which the tangents drawn to the circle $x^{2}+y^{2}=25$ are perpendicular, are
(A) $(\sqrt{46}, 2)$
(B) $(-\sqrt{46}, 2)$
(C) $(\sqrt{37}, 2)$
(D) $(-\sqrt{37}, 2)$

Key Concepts

Circle Geometry

This concept involves understanding the equation of a circle, its center, and its radius. The circle’s equation in standard form provides key information about its geometric properties, which are critical when studying the relationships between points, lines, and curves in the coordinate plane.

Tangents to a Circle

Tangents are lines that touch the circle at exactly one point. The properties of tangents, such as their equal lengths from an external point to their points of contact, are instrumental in solving problems involving circles and lines, especially when determining conditions for tangency and the existence of these tangents from a given external point.

Perpendicular Tangents Condition

When two tangents are drawn from an external point to a circle, they form an angle at that point. A specific condition arises when these tangents are perpendicular to each other. This condition relates the distance from the external point to the center of the circle with the radius of the circle, often leading to a relationship where the square of the distance equals twice the square of the radius.

Distance Formula in Coordinate Geometry

The distance formula is used to calculate the distance between two points in the coordinate plane. It is vital for finding distances such as between an external point and the center of a circle, which is necessary when applying conditions like those that determine when tangents from a point will be perpendicular.

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