Danica Mona and Hiro are having a discussion about parabolas

Danica, Mona, and Hiro are having a discussion about parabolas and circles. Mona states that she thinks that if the center of a circle lies on a parabola and the circle passes through the focus of that parabola, then the directrix of the parabola must be tangent to the circle. Hiro responds that though this is sometimes true, there are circles that have a center on a parabola and pass through the focus, but the directrix is not tangent to them. Danica says she doesn\'t think that this is ever true. Determine which member of the discussion is correct. Your response will include two parts: You need to create two different detailed graphs that support your assertion. The graphs should be created by hand and be clearly labeled with all relevant information including the corresponding equations. The equations for the parabola and the circle in the second example should be completely different from the equations used in the first example. If you agree with Mona, then you should provide two distinct examples that support her statement. If you agree with Hiro, then you should provide one example that shows Mona\'s statement to be true and one counterexample. If you agree with Danica, then you should show two distinct counterexamples that illustrate that Mona\'s statement is false. In a detailed paragraph, explain your reasoning. Be sure to use relevant mathematical vocabulary and proper conventions of writing. You should be sure to incorporate the precise definitions of a circle and a parabola. Refer to your diagrams as a means to support your assertion.

Solution

To solve the problem let us assume that assertion of any of Mona ,Hiro or Danica is right.

For discussion sake ,let us assume that Mona is speaking the truth .

\"if a center of a circle lies on a given paralabola and passes through its focus then the directrix of the parabola must be tangent to the circle \"

Let us take a standard parabola of the form

y² =4ax ;

Clearly for this parabola, the focus will be F(a,0) and the directrix equation will be x=-a .

Now let us take a general equation of a circle

x²+y²+2gx+2fy+c=0   . clearly its center will be (-g,-f) .

Now it passes through the focus F(a,0) of the parabola

=> a²+0+2ga+0+c=0 (keeping values of F in circle equation)

a²+2ag+c=0 ...

Also the center of the circle lies on the parabola hence its focal distance which will also be its radius will be gven by

r=(a+x) for any point (x,y) on parabola .

Clearly

since we have assumed that the circle passes through the directrix line and is tangent to it .

=> it passes through (-a,0) ...... iii

From all the three assumptions we can infer that

the center of the circle will be at (0,0)

But (0,0) is origin and the parabola that we have taken passes through it.

Thus this re-validates our assumption that the center of the circle lies on the parabola and it is at origin.

Thus we can say that

For any given standard parabola , if the center of the circle passes through it and the focus of the parabola lies on the circle then the directrix of the parabola will always be tangent to circle and the center of the circle will lie at the origin \"

Hence from our analysis we can say that Hiro is correct .

i.e, if the center of the circle passes through it and the focus of the parabola lies on the circle then the directrix of the parabola will be tangent to circle only sometimes and it will happen only when the center of the circle will lie at the origin .