Find equation of circles using parabola

Question

Parabola $y=\frac{1}{16}x^2-\frac{3}{4}x+\frac{25}{4}$

A line passing through the origin and point $(6,8)$ and $x$-axis are tangents to the circles that I'm supposed to find.

How can I find the equation of the circles using this information and the given parabola?

Answer 1

The general equation of a circle with center $(a,b)$ and radius $r$ is given by $$ (x-a)^2 + (y-b)^2 = r^2$$ The two lines that should be tangent to the circle are given by \begin{align} y &= \frac{4}{3}x\\ y &= 0\\ \end{align} For the circle to be tangent to these lines the center of the circle should be in the middle of these lines, to be precise on the line $$y = \frac{2}{3}x$$ And thus the radius is $\frac{2}{3}a$. The general equation (before taking the parabola into account) is $$ (x-a)^2 + (y-\frac{2}{3}a)^2 = \left(\frac{2}{3}a\right)^2 $$ We now use the parabola to find the last constrait on $a$

Answer 2

Sorry to say that your problem is hard to understand. May it be the following?

Problem

Find the equation of the circle such that

1. the center of the circle lies on the parabola $y=\dfrac{1}{16}x^2-\dfrac{3}{4}x+\dfrac{25}{4}$;
2. the circle is tangent to the line passing through $(0,0),(6,8)$ and the $x-$axis.