#### To determine

**To Find:** The center of the circle.

#### Answer

Center of the circle is

0,54.

#### Explanation

**1) Concept:**

Use given information and equation of circle with center (a, b) and radius r.

From the given figure circle touches parabola at two points. These points satisfy equation of the circle.

We can use this to find center of circle.

Another way to solve the problem is to notice that the at point where they touch the circle and parabola has same tangent. We can solve for c by equating the slope of the tangent.

**2) Formula:**

Equation of circle with center (a, b) and radius r is given by,x-a2+y-b2=r2

**3) Given:**

Circle with radius 1 is inscribed in parabola y=x2.

**4) Calculations:**

From given figure center of circle is (0, c) where c is y-intercept.

Equation of circle with center (0, c) and radius 1 is given by,

x2+y-c2=1

As parabola intersects circle at two points, therefore, equation of parabola y= x2 satisfy equation of circle.

Using y=x2 in equation of circle,

y+y-c2=1

y+y2-2cy+c2=1

y2-2c-1y+c2-1=0

Parabola intersects circle at two points but from figure y- coordinates of these points are same.

Therefore, discriminant of above polynomial is 0.

Hence 2c-12-4c2-1=0

4c2-4c+1-4c2+4=0

-4c+5=0

c=54

**Alternate method:**

Let (a, b) be a common point of the circle and tangent. Consider the common tangent at this point. We can find the slope of this tangent by evaluating the derivative of either of the curve at x=a. Let this slope be m.

Equation of parabola is y=x2. Differentiating it we get

dydx=2x

Thus the slope of tangent at (a, b)

m =2a

Now the line is tangent to the circle as well. Let us use implicit differentiation to obtain an expression for slope from the circle. The equation of the circle is

x2+y-c2=1.

Differentiating both sides we have

2x+2(y-c)dydx=0

This implies

dydx=-x(y-c)

Now since the line is tangent to the circle as well the value of above expression at x=a shall give slope of the line. At x = a since the point is also on the parabola y = a2 Thus we have

-a(a2-c)=m=2a

Thus we have 2(a2-c)=-1.

From which we can conclude that c=a2+12

Now (0,c=a2+12) is the center of the circle and (a,a2) is a point on the circle. The radius of the circle is 1. So we have

(0-a)2+(a2+12-a2)2=1

That is

a2+14=1

So a2=34. Substituting this value in c=a2+12 we see that

c=54

So the centre of circle is

0,c=0,54

**Conclusion:**

Center of the circle is:

0,54