Circle Tangent to a Parabola?

Geometry Level 5

The above picture shows the graphs of the parabola $y = x^{2}$ and the line $y = -\frac{1}{2}x$ in Cartesian coordinates. A circle with center $(p, 0)$ , where $p > 0$ , is tangent to both the parabola and the line. What is the value of $p$ rounded to three decimal places?

Graph generated in Desmos .

The answer is 0.677.

1 solution

Ronak Agarwal
Oct 25, 2014

We have the centre of the circle as $A=(x,0)$ let the point of contact of circle and parabola be $B=(t,{t}^{2})$ .

Ist equation we would be writing is that the tangent and AB is perpendicular to each other, hence we write :

$(Slope\quad of \quad tangent)(Slope\quad AB)=-1$

$(2t)(\frac{{t}^{2}}{t-x})=-1$

rewriting it we have :

$2{t}^{3}+t=x$ Equation 1

Second equation we would be writing is that :

length(AB)=Perpendicular distance from the line $x+2y=0$

$\Rightarrow {t}^{4}+{(t-x)}^{2}=\frac{{x}^{2}}{5}$ Equation 2

Solving these two equations we get :

$\large x=\frac{(15+\sqrt{65})(\sqrt{2\sqrt{65}-2})}{128}$

Hello Ronak Agarwal, that's a very good solution. However, I also tried solving it using a different approach and I got the final answer as $\frac{\sqrt{830+130\sqrt{65}}}{64}$ which has the same value as your answer but in a different form. Can you please guide me on how to convert my answer into this form? $\frac{(a + \sqrt{b})(\sqrt{-c+d\sqrt{e}})}{f}$ Thank you very much.

Daren Wong - 5 years, 6 months ago

actually the question has been wrongly asked i did get same and also converted it but the answer i received was different check this (8+sqrt(65))(sqrt(-28130+3490sqrt(65))/64

aryan goyat - 5 years, 3 months ago

Circle Tangent to a Parabola?

The answer is 0.677.

1 solution

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