The figure shows a fixed circle C? with equation (x ? 1)" + y" = 1 and a shrinking circle C? wit

Step 1: Understand the given information. We have two circles, C1 with the equation (x^2 + y^2

The figure shows a fixed circle C? with equation (x ? 1)" +...

The figure shows a fixed circle C? with equation (x ? 1)" + y" = 1 and a shrinking circle C? with radius r and center the origin. P is the point (0, r), Q is the upper point of the intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to the x-coordinate of R as C? shrinks, that is, as r ? 0?? The x-coordinate of R ? -1 If the answer is infinity or negative infinity, enter I or ?I as appropriate. If the limit does not exist, type DNE.

Transcript

00:01 Hi, in this question we have been given two circles c1 is a fixed circle and c2 is a circle that is shrinking.

00:06 The radius r has been given of c2.

00:09 Now the point of p has been given to us as 0 comma r and the point q is the point of intersection of the two circles.

00:19 We have been asked if r tends to 0 then what will be the coordinates of the point are.

00:25 Means if radius tends to 0 what will be the coordinates of the point are and the center of c2 is the origin so first let's find the equation of the line pq for that we have the coordinates of point p let us find the coordinates of point 2 so now for that we can say that q it is the point of intersection of two circles and the point of intersection of two circles and the the first equation of the circle that is of c1 circle is given to us as this.

01:08 Now for the second, for the c2 circle we can say that the center is 0 .0 and the radius it is r.

01:23 So thus we will have equation as x square plus y square that is equal to r square.

01:29 Let us name this as equation number 2.

01:32 We will be solving the equation.

01:34 Equation 1 and equation number 2 simultaneously, that is x minus 1 the whole square plus y square is equal to 1 and x square plus y square it is equal to r square.

01:47 Subracting the equation signs will change.

01:51 So does we will be getting this as minus 2x which is equal to minus r square.

01:57 So therefore we will say that x be equal to r squared divided by 2.

02:02 Now we will substitute x which is equal to r squared divided by 2 in the equation number 2.

02:12 So does we get this as y square which is equal to r square minus x square as we have to find the point of y now? so does this will be equal to r square into 1 minus r squared by 4 and therefore y will be equal to square root of 1 minus r squared.

02:32 Minus r squared divided by 4 into r so thus these are the points of x and y which we have received so therefore we can say that the point q it is equal to r squared divided by 2 comma r into square root of 1 minus r squared divided by 4 further now to find the equation of the line we will be using a two point form that is y minus 1 y1 is equal to y2 minus y1 divided by x2 minus x1 into x minus x1, which this can be written as y minus y1 divided by x minus x1.

03:18 Thus, this is equal to y2 minus y1 divided by x2 minus x1.

03:24 So the point of p and q we have substitute the points that as we will get this as y minus r divided by x minus 0.

03:33 This will be equal to r into square root of 1 minus r squared divided by 4 minus r, the whole divided by r squared by 4 minus 0...

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