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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?

Kenta S
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upandcomer
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  • What is a "line passing through the origin and point (6,8) and x-axis" ??? And where does the parabola come into play ? –  Jul 05 '18 at 10:14
  • How is the parabola relevant here? Does it pass through the circle center? – Dylan Jul 05 '18 at 11:16
  • Not very polite to ignore the comments. –  Jul 05 '18 at 11:22

2 Answers2

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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 $a$

Tim Dikland
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    I am always amazed to see how people are able to answer completely unintelligible questions. –  Jul 05 '18 at 10:13
  • You are right, I am assuming a question. – Tim Dikland Jul 05 '18 at 10:22
  • "We now use the parabola to find the last constraint on $a$" ... I suppose that's the prudent place to stop, since the problem statement doesn't explain how to use the parabola. (Is the circle centered on the parabola? tangent to it? Does it pass through the focus? the vertex? something else? Who knows?) – Blue Jul 05 '18 at 10:43
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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.

enter image description here

Community
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mengdie1982
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  • If that's not the problem statement, it should be ... although it's a bit of a shame to have the exercise presented as an algebraic slog. There's some really nice geometry going on here (as your image suggests you already know). $(6,8)$ isn't just some random point: it's the parabola's focus. Moreover, the $x$-axis is its directrix. The two angle bisectors are naturally perpendicular, and they're tangent to the parabola at the circles' centers, which in turn define a focal chord, which itself is perpendicular to the line through the origin and $(6,8)$. Neat stuff! – Blue Jul 05 '18 at 11:09