Locus examples

Adrian

2010-11-11 12:12 am

Let A and B be two fixed points. P is a moving point.
What is the locus (geometric meaning, eg circle, ellipse...) of(1) PA = 2 PB(2) 3PA = 2PB(3) PA + PB = 2(4) PA - PB = 2Let F be a fixed point. P is a moving point.
(5) What is the locus (geometric meaning, eg circle, ellipise...) of
distance between P and F equal to distance between P and a fixed straight line?(6) Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, 點樣由 coefficient 睇到條 equation 係 conics 裡的 circle, parabola, ellipse or hyperbola?

回答 (1)

用戶25975

2010-11-11 12:39 am

✔ 最佳答案

(1) and (2): Circle

For any positive value of k NOT equal to 1, when PA = k PB, the locus describes a circle

(3): Ellipse

Ellipse is a locus of moving point so that the SUM of its distances from two fixed points, known as foci, is a constant.

(4): Hyperbola

Hyperbola is a locus of moving point so that the DIFFERENCE of its distances from two fixed points, known as foci, is a constant.

(5): Parabola

Parabola is a locus of moving point so that the distance from it to a fixed point (known as focus) is always equal to that from it to a fixed line (known as directrix)

(6) Obviously when C = 0 and A = B, it will be a circle

Then for other situations, we have to check the sign (positive or negative) of the value AB - (C/2)^2.

If AB - (C/2)^2 = 0, then it is a parabola

If AB - (C/2)^2 > 0, then it is an ellipse

If AB - (C/2)^2 < 0, then it is a hyperbola

參考: 原創答案

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