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September 4th, 2017, 08:44 AM   #1
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Find the relation between (a) and (b)

Hi everyone,

If the line y = ax+b is tangential to the unit circle, then find the relation between (a) and (b).

Any help please?
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September 4th, 2017, 09:17 AM   #2
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The unit circle has the equation $x^2 + y^2 = 1$. You know that the line touches the circle in only one distinct point.

At the point of intersection, both equations of the system \begin{align*}y &= ax + b \\ x^2 + y^2 = 1\end{align*}
are satisfied. You can thus solve the system for $x$ (or y, but $x$ looks easier to me), and then knowing that there is exactly one such point (a repeated root), you will be able to determine a condition for $a$ and $b$.
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September 4th, 2017, 09:25 AM   #3
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Quote:
Originally Posted by v8archie View Post
The unit circle has the equation $x^2 + y^2 = 1$. You know that the line touches the circle in only one distinct point.

At the point of intersection, both equations of the system \begin{align*}y &= ax + b \\ x^2 + y^2 = 1\end{align*}
are satisfied. You can thus solve the system for $x$ (or y, but $x$ looks easier to me), and then knowing that there is exactly one such point (a repeated root), you will be able to determine a condition for $a$ and $b$.
I already tried that, but I could not find the relation between (a) and (b)
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September 4th, 2017, 09:57 AM   #4
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So you have something like:
$$y = ax + b \implies y^2 = (ax+b)^2$$
and so
\begin{align*}x^2 + y^2 &= 1 & \implies x^2 + (ax+b)^2 &= 1 \\
&& x^2 + a^2x^2 + 2abx + b^2 -1 &= 0 \\
&& (a^2+1)x^2 + 2abx + (b^2 - 1) &= 0\end{align*}

Now, this is a quadratic equation in $x$. We know that we must have only one (distinct) solution. What condition on the coefficients of a quadratic equation holds when there is only one solution (i.e. it's a repeated root)? And what does this condition mean in this particular case?

Last edited by v8archie; September 4th, 2017 at 10:00 AM.
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September 6th, 2017, 04:55 PM   #5
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Does the repeated root mean that a = b? Does it mean that a + b = 0?
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September 6th, 2017, 06:41 PM   #6
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No. The general quadratic equation $Ax^2 + Bx + C = 0$ has a repeated root when $B^2 = 4AC$.
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