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September 4th, 2017, 07:44 AM  #1 
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1  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? 
September 4th, 2017, 08:17 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
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 $b$. 
September 4th, 2017, 08:25 AM  #3  
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1  Quote:
 
September 4th, 2017, 08:57 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
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 09:00 AM. 
September 6th, 2017, 03:55 PM  #5 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 661 Thanks: 87 
Does the repeated root mean that a = b? Does it mean that a + b = 0?

September 6th, 2017, 05:41 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
No. The general quadratic equation $Ax^2 + Bx + C = 0$ has a repeated root when $B^2 = 4AC$.


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