Loci：練習問題

A point \(P(x, y)\) moves such that it maintains a constant distance of \(6\) from the straight line \(y = -2\). Which of the following is an equation representing the locus of \(P\)?

A point P(\(x, y\)) moves such that its distance from the fixed point F(\(4, 0\)) is always twice its distance from the vertical line \(L: x = 1\). Find the equation of the locus of P.

Find the equation of the locus of a point \(P(x, y)\) such that its distance from the point \(A(4, 0)\) is always half of its distance from the point \(B(-2, 0)\).

A point \(P(x, y)\) moves such that its distance from the line \(L_1: 4x + 3y - 5 = 0\) is always equal to its distance from the line \(L_2: 4x + 3y + 15 = 0\). Which of the following is the equation of the locus of \(P\)?

The coordinates of the points \(A\) and \(B\) are \((-1, 3)\) and \((5, 1)\) respectively. A point \(P(x, y)\) moves in the rectangular coordinate plane such that \(AP^2 + BP^2 = 30\). Find the equation of the locus of \(P\).

Consider a point \(P(x, y)\) that moves in a Cartesian plane such that its distance from the line \(L: y = 4\) is always 3 units.

(a) Find the equations of the locus of \(P\).
(b) Describe the locus of \(P\) geometrically.
(c) If the point \(P\) also lies on the line \(x = 2\), find all possible coordinates of \(P\).

Consider two parallel lines \(L_1: 3x - 4y + 2 = 0\) and \(L_2: 3x - 4y - 18 = 0\).

(a) Find the equation of the locus of a point \(P(x, y)\) which is always equidistant from \(L_1\) and \(L_2\).
(b) Find the perpendicular distance between the locus found in (a) and the line \(L_1\).