The center of a circle is at (-3,-2) and its radius is 7. Find the length of the chord which is bisected at (3,1). At what points does the circle cut the y - axis?

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Given a circle centered at (-3,-2) with radius 7:

(1) Let the center of the circle be C. Let `bar(AB)` be a chord of the circle such that its midpoint M is at (3,1). Find `AB` :

Recall that a radius drawn to the midpoint of a chord is perpendicular to the chord. (Two points that are equidistant from the endpoints of a segment lie on the perpendicular bisector of the segment.)

We find CM using the distance formula: `CM=sqrt((-3-3)^2+(-2-1)^2)=sqrt(36+9)=sqrt(45)=3sqrt(5)`

Note that CA=7 (it is a radius) and that `Delta CMA` is a right triangle. Using the Pythagorean theorem we find AM:





Since M is the midpoint of `bar(AB)` , we have AB=4.


The length of the chord bisected at (3,1) of the given circle is 4.


(2) To find the y-intercepts of the circle, we write its equation in standard form:


The y-intercepts occur when x=0, so:







The y-intercepts of the circle are at `-2+2sqrt(10),-2-2sqrt(10)` or approximately `4.32,-8.32`


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