The common tangents to the circle x^2+y^2=2 and the parabola...

The common tangents to the circle $x^{2}+y^{2}=2$ and the parabola $y^{2}=8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then, the area (in sq units) of the quadrilateral $P Q R S$ is

Key Concepts

Tangent Lines to Conic Sections

A tangent line to a conic section touches the curve at exactly one point, meaning that when the line's equation is substituted into the conic's equation, the resulting equation has exactly one solution. This concept is used across different conics, such as circles and parabolas, by applying algebraic conditions that ensure discriminants vanish, characterizing the tangency condition.

Common Tangents

Common tangents are lines that are simultaneous tangents to two or more curves. In problems involving conic sections, finding common tangents involves setting the tangency conditions for each curve and solving the resulting system, thereby identifying any line that touches each conic at exactly one point. This technique is fundamental in understanding the mutual geometry of multiple curves.

Condition for Tangency to a Circle

For a circle, the condition for a line to be tangent is that the perpendicular distance from the center of the circle to the line equals the radius. This geometric condition can be translated into an algebraic equation involving the line's slope and intercept, and the circle's center and radius, enabling systematic determination of tangent lines in coordinate geometry.

Condition for Tangency to a Parabola

A tangent to a parabola only touches it at one point, and the condition for tangency involves ensuring that the line and the parabola intersect at a single point. In algebraic terms, when substituting the line's equation into the parabola's equation, the resulting quadratic equation in the coordinate variable must have a discriminant of zero. This yields specific relationships between the slope and intercept of the tangent line relative to the parabola.

Analytic Geometry in Area Calculation

Once the points of tangency or intersection between curves are determined, the area of the resulting polygon, such as a quadrilateral, can be calculated using techniques from analytic geometry. Methods like the shoelace formula or decomposition into simpler geometric figures are commonly employed to find the area based on the coordinates of the vertices.

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