A line passing through P(3,5) intersects the circle x^2+y^2=4 at A and B. A point Q is taken on

step 1 In this problem of conic section, we have given that a line passing through the point P whose co

A line passing through P(3,5) intersects the circle...

A line passing through $\mathrm{P}(3,5)$ intersects the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=4$ at $\mathrm{A}$ and $\mathrm{B}$. A point $\mathrm{Q}$ is taken on $\mathrm{AB}$ such that $2 \mathrm{PQ}=$ $\mathrm{PA}+\mathrm{PB}$. Then, the locus of $\mathrm{Q}$ is (a) $x^{2}+y^{2}-3 x+5 y=0$ (b) $x^{2}+y^{2}+3 x-5 y=0$ (c) $x^{2}+y^{2}+3 x+5 y=0$ (d) $x^{2}+y^{2}-3 x-5 y=0$

A line passing through $\mathrm{P}(3,5)$ intersects the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=4$ at $\mathrm{A}$ and $\mathrm{B}$. A point $\mathrm{Q}$ is taken on $\mathrm{AB}$ such that $2 \mathrm{PQ}=$ $\mathrm{PA}+\mathrm{PB}$. Then, the locus of $\mathrm{Q}$ is
(a) $x^{2}+y^{2}-3 x+5 y=0$
(b) $x^{2}+y^{2}+3 x-5 y=0$
(c) $x^{2}+y^{2}+3 x+5 y=0$
(d) $x^{2}+y^{2}-3 x-5 y=0$

Key Concepts

Algebraic Techniques in Locus Problems

Problems that ask for the locus of a point defined by a geometric condition often require eliminating parameters (such as the slope of a secant) to obtain an equation in x and y. This involves algebraic manipulation, completing the square, and sometimes leveraging symmetric properties of the geometric configuration. Mastery of these techniques is essential for translating geometric relationships into their algebraic counterparts.

Circle Equation

A circle in the coordinate plane is typically given by an algebraic equation of the form x² + y² + Dx + Ey + F = 0. This form encapsulates the geometric properties of the circle such as its center and radius, which can be determined by completing the square. Understanding the circle’s equation is fundamental in problems where intersections, chords, or loci relative to the circle are studied.

Locus

The concept of a locus involves finding the set of points that satisfy a specific geometric condition or equation. In problems where a point moves according to a given rule (for example, how Q is chosen relative to a chord), determining its locus means deriving a fixed equation that every such point obeys. This idea is central to coordinate geometry and is used to convert geometric constraints into algebraic equations.

Chord of a Circle

A chord is a line segment whose endpoints lie on the circle. When a line passes through a point (which may lie outside the circle) and intersects the circle, it forms a chord. The properties of chords—such as their lengths, midpoints, and the relationships between them—play a crucial role in many circle geometry problems, especially those dealing with distances and division ratios along the chord.

Secant Line and External Point Relations

When a line (called a secant) passes through an external point and cuts a circle at two points, certain distance relations arise. One key relation is the power of a point theorem, which involves the products of distances from the external point to the intersection points. In problems where additional conditions (like a relation between distances on the chord and from the external point) are imposed, these relations help in establishing the position of a point on the chord whose location is independent of the specific secant chosen.

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