(i) Find the equations of the common tangents of the circle (x-3)^2+y^2=9 and y^2=4 x. (ii) Find

step 1 In this problem of conic section, we have to find the equation of common tangent of the circle w

(i) Find the equations of the common tangents of the circle...

Key Concepts

Common Tangents

Common tangents are lines that are tangent to two or more curves simultaneously. To find common tangents to different conic sections, one must derive conditions that satisfy the tangency condition for each curve and then solve these conditions concurrently. This process often involves equating parameters obtained from the tangency conditions of each curve.

Parabola Geometry

A parabola is a set of points equidistant from a fixed point (the focus) and a line (the directrix). Its standard equation provides information about its orientation, vertex, and axis of symmetry. Understanding these properties is key to analyzing tangency conditions and interactions with other curves, such as identifying where a line may touch the parabola.

Area of Bounded Regions

Once the common tangents are determined, the region bounded by these lines can be analyzed using coordinate geometry. The vertices of the enclosed polygon, which arise as the intersection points of the tangent lines, can be used to compute the area. Methods such as the shoelace formula or integrating the difference between functions may be applied to determine the area of the bounded region.

Discriminant Condition for Tangency

When substituting the equation of a line into the equation of a conic section, a quadratic equation in one variable is obtained. The line is tangent to the conic if and only if the discriminant of this quadratic equation is zero. This condition guarantees that the equation has exactly one real solution, corresponding to the point where the line touches the curve.

Tangent to a Conic Section

A tangent line to a conic section is a straight line that touches the curve at exactly one point, and this point of contact represents the point where the line and the curve have the same slope. The condition for tangency is typically established by ensuring that when the line's equation is substituted into the conic's equation, the resulting quadratic has a single repeated solution, indicating only one point of intersection.

Circle Geometry

A circle is a set of points in a plane that are equidistant from a fixed center point. The standard equation of a circle, (x - h)² + (y - k)² = r², indicates a circle with center (h, k) and radius r. Knowledge of circle geometry, including its center and radius, is crucial for finding tangent lines and understanding relative positions to other curves.

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