Given x^2+y^2=8 x represents the circle C and y^2=4 x...

Given $x^{2}+y^{2}=8 x$ represents the circle $C$ and $y^{2}=4 x$ represents the parabola $P$. Then (a) $(2,3)$ is point is region bounded by $\mathrm{C}$ and $\mathrm{P}$ (b) $(2,-3)$ is a point inside of both $\mathrm{C}$ and $\mathrm{P}$ (c) $(2,4)$ is a point outside of both $\mathrm{C}$ and $\mathrm{P}$ (d) $(2,-4)$ is a point is a region bounded by $C$ and $P$

Given $x^{2}+y^{2}=8 x$ represents the circle $C$ and $y^{2}=4 x$ represents the parabola $P$. Then
(a) $(2,3)$ is point is region bounded by $\mathrm{C}$ and $\mathrm{P}$
(b) $(2,-3)$ is a point inside of both $\mathrm{C}$ and $\mathrm{P}$
(c) $(2,4)$ is a point outside of both $\mathrm{C}$ and $\mathrm{P}$
(d) $(2,-4)$ is a point is a region bounded by $C$ and $P$

Key Concepts

Point-curve Positional Testing

This key concept entails substituting the coordinates of points into the equations of curves to determine their relative positions—whether a point lies inside, on, or outside a given curve. For circles, it is done by comparing the squared distance from the center to the radius squared; for parabolas, it requires evaluating the sign of the difference when the point’s coordinate is substituted into the equation. This method is a central technique in coordinate geometry for assessing set inclusion and analyzing regions defined by curves.

Regions Defined by Multiple Curves

This concept relates to understanding and delineating areas in the plane that are bounded by more than one curve. It involves studying the intersections and relative positions of different curves, such as circles and parabolas, to define regions that are common to them or lie between them. This analysis is crucial for solving problems involving composite shapes and regions in analytic geometry, as it leads to an understanding of how different curves partition the plane.

Circle Equation in Standard Form

This concept involves rewriting the given circle’s equation by completing the square in order to identify its center and radius. Doing so transforms the equation into a form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r its radius. This conversion is fundamental in analytic geometry as it simplifies the identification of the circle’s geometric properties and facilitates the analysis of positional relationships with other curves or points.

Parabola Equation in Standard Form

This concept refers to manipulating the equation of a parabola to express it in a form that reveals its orientation, vertex, and focal properties. For parabolas opening sideways or vertically, writing the equation in standard form, such as y²=4ax or x²=4ay, helps to identify the direction of opening and the geometric features. Such transformations are important for comparing the parabola with other curves and for analyzing intersections or relative positions of points.

Please give Ace some feedback