Tiêu đề 61 Conic Sections - Circles.pdf ✅

MSTC 61: Conic Sections – Circles 1. Locus A circle is the locus of points equidistant to a point called the center. For a point (x, y) centered at (h, k) with radius r, using the distance formula, r = √(x − h) 2 + (y − k) 2 (x − h) 2 + (y − k) 2 = r 2 This equation is called the center-radius form of the circle. Expanding, x 2 − 2hx + h 2 + y 2 − 2yk + k 2 = r 2 x 2 + y 2 − 2hx − 2ky + h 2 + k 2 − r 2 = 0 Let D = −2h, E = −2k, and F = h 2 + k 2 − r 2 , x 2 + y 2 + Dx + Ey + F = 0 This form is called the standard equation of a circle. 2. Radical Axis The radical axis of a pair of circles is the perpendicular bisector of the segment connecting the centers of the circles. If the circles also intersect each other, then the radical axis passes through the points of intersection. The equation of the radical axis can be found by eliminating the square terms in the standard equations of the circle. For example, What is the equation of the radical axis of x 2 + y 2 − 2x = 0 and 3x 2 + 3y 2 − 3x − 6y + 10 = 0? Eliminating the square terms by subtracting thrice the first equation from the second equation, (3x 2 + 3y 2 − 3x − 6y + 10) − 3(x 2 + y 2 − 2x) = 0 − 3(0) −9x − 6y + 10 = 0 9x + 6y − 10 = 0