The coordinates of any point on the circle through the points A(2,2), B(5,3) and C(3,-1) can be

step 1 In this problem of conic section, we have given that the coordinates of any point on the circle

The coordinates of any point on the circle through the...

The coordinates of any point on the circle through the points $\mathrm{A}(2,2), \mathrm{B}(5,3)$ and $\mathrm{C}(3,-1)$ can be written in the form $(4+\sqrt{5} \cos \theta, 1+\sqrt{5} \sin \theta) .$ Then, the coordinates of the point $\mathrm{P}$ on $\mathrm{BC}$ such that $\mathrm{AP}$ is perpendicular to $\mathrm{BC}$ are (a) $(-1,4)$ (b) $(4,1)$ (c) $(1,4)$ (d) $(2,3)$

Key Concepts

Foot of the Perpendicular

The foot of the perpendicular from a point to a line is the point on the line that is closest to the external point, and it can be determined using vector projection techniques or algebraic methods involving slopes. This concept is used to solve problems where a segment drawn from a point must be perpendicular to a given line, ensuring optimal distances or orthogonal intersections.

Equation of a Line from Two Points

The equation of a line through two given points can be determined using the two-point form or by calculating the slope and using point-slope form. This concept is fundamental when working with line segments and their properties, such as finding intersections or projections related to circles or other geometric figures.

Circle Equation and Parametric Representation

A circle in the coordinate plane is typically described by its center and radius. The general equation (x - h)² + (y - k)² = r² represents all points equidistant from the center (h, k). When expressed parametrically, the coordinates can be written as (h + r cos?, k + r sin?), which is useful for analyzing points on the circle using an angle parameter.

Perpendicularity in Coordinate Geometry

In coordinate geometry, two lines are perpendicular if the product of their slopes is -1, or equivalently, if their direction vectors have a dot product of zero. This concept is crucial when determining conditions for orthogonality, such as finding a line or segment that is perpendicular to a given line.

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