Tangents drawn to x^2+y^2=16 from P(0, h) meet the x -axis...

Tangents drawn to $x^{2}+y^{2}=16$ from $P(0, h)$ meet the $x$ -axis at $A$ and $B$. The value of $h$ for which the area of triangle $\mathrm{PAB}$ is minimum is
(a) 32
(b) $\sqrt{30}$
(c) $\sqrt{31}$
(d) $4 \sqrt{2}$

Key Concepts

Optimization Techniques in Geometry

Optimization in a geometric context involves finding the maximum or minimum values of a particular quantity, such as area or distance. This usually requires setting up an algebraic expression for the quantity in question and then applying calculus—taking derivatives and setting them to zero—to identify the extremum conditions.

Triangle Area Computation in Coordinate Geometry

Calculating the area of a triangle using coordinates typically involves using formulas based on the vertices’ positions. In coordinate geometry, determining the area of a triangle formed by intersecting lines (such as tangents) and the axes involves translating geometric configurations into algebraic expressions.

Tangent Lines to Circles

A tangent line to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact. Understanding the conditions under which a line is tangent to a circle—often by using the discriminant method for the intersection of a line and a circle—is essential for problems that involve drawing tangents from an external point.

Circle Geometry

Circle geometry involves understanding the fundamental properties of circles, including the concept of radius, center, and the standard equation of a circle. It provides the foundation for analyzing problems that involve relationships between points and circles, such as determining if a line or point lies on, inside, or outside the circle.

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