Find the equation of a circle circumscribing the segment of a parabola
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y2 = 4ax cut off by the latus rectum of the parabola.
Answers
Answer:
A circle is completely determined by giving three points, so you need two steps to do this: first compute the three points where your circle passes (the tip of the parabola and the endpoints of your segment presumably, if that is what you mean by “circumscribing a segment”, which usually would just mean passing through the endpoints”), and secondly computing the circle equation from those three points.
Assuming that a is not equal to zero, the endpoint of your parabola is the origin (0,0), the focus is at (4a,0), the latus rectum is the vertical line of equation x=4a, and it cuts the parabola at the two points (4a,4a) and (4a,-4a). So the circle should pass through (0,0), (4a,4a) and (4a,-4a). That settles step one.
To find the equation of the circle passing through these three points, find the centerpoint of the circle (p,q) that is equidistant from all three points. This means that for some distance r (the radius), we have r^2 = p^2+q^2 = (4a-p)^2+(4a-q)^2 = (4a-p)^2+(-4a-q)^2.
The third equality gives (4a-q)^2=(-4a-q)^2, which expands out to 16a^2–8aq+q^2=16a^2+8aq+q^2, so we have -8aq=8aq, so since we assumed that a is not zero, we find that q=0.
Plugging zero into the second equality gives p^2=(4a-p)^2+16a^2, which expands out to p^2=p^2–8ap+32a^2, so 8ap=32a^2, so p=4a.
So the center of the circle is at (p,q)=(4a,0). The distance of this point from (0,0) is 4a, which is the radius, so the circle equation is
(x-4a)^2+y^2=16a^2.
Step-by-step explanation:
Answer:
x^2 +y^2 - 10x=0
Step-by-step explanation:
so u can clearly see process in these photos... tq
