User Name Remember Me? Password

 Geometry Geometry Math Forum

 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: 7,690 Thanks: 2669 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:
 Originally Posted by v8archie 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) September 4th, 2017, 08:57 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 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: 673 Thanks: 88 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: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra No. The general quadratic equation $Ax^2 + Bx + C = 0$ has a repeated root when $B^2 = 4AC$. Thanks from Benit13 Tags find, relation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post siri Math 3 June 17th, 2017 12:03 PM dhairya bhardwaj Math 3 September 8th, 2016 04:59 AM study7 Number Theory 0 November 12th, 2014 03:54 AM finalight Applied Math 13 October 14th, 2011 09:16 AM robocop_911 Applied Math 0 June 4th, 2008 11:40 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      