My Math Forum  

Go Back   My Math Forum > High School Math Forum > Trigonometry

Trigonometry Trigonometry Math Forum


Reply
 
LinkBack Thread Tools Display Modes
September 9th, 2008, 10:17 AM   #1
Lax
Newbie
 
Joined: Sep 2008

Posts: 1
Thanks: 0

A problem concerning a tangent line and a circle

First of all hi to everybody, I'm new to these forums, but i was previously quite active on another math-related forum (which is sadly almost dead nowadays).

So here's the problem I currently can't solve,

For what values of k is the straight line y = kx + 1 a tangent to the circle with the origin at (5, 1) and a radius of 3.
Lax is offline  
 
September 9th, 2008, 02:45 PM   #2
Senior Member
 
Joined: May 2008
From: York, UK

Posts: 1,300
Thanks: 0

Re: A problem concerning a tangent line and a circle

A circle of radius r, centre $\displaystyle (x_0,y_0)$ can be described by the equation $\displaystyle (x-x_0)^2+(y-y_0)^2=r^2.$

When you have simplified this for the circle given in the problem, you need to find values of x, y and k such that

(i) The line and the circle intersect (i.e. x, y and k satisfy both the equation for the line and the equation for the circle);

(ii) The slope of the line and the circle are equal at this point (i.e. $\displaystyle \frac{dy}{dx}$ is equal on the line and the circle when evaluated for these values of x, y and k.)

Last edited by skipjack; February 28th, 2018 at 12:57 PM.
mattpi is offline  
September 9th, 2008, 04:56 PM   #3
Senior Member
 
Joined: Jul 2008

Posts: 895
Thanks: 0

Re: A problem concerning a tangent line and a circle

If not differentiating yet, the second relation can be found from the fact that the line from the point of tangency to the center is perpendicular to the tangent line y = kx + 1.
Dave is offline  
September 10th, 2008, 01:02 PM   #4
Global Moderator
 
Joined: May 2007

Posts: 6,805
Thanks: 716

Re: A problem concerning a tangent line and a circle

Compute the intersection of the line (as a function of k) with the circle. The solution set will consist of one of the following: two different intersections, two identical intersections, or no intersection. The k's (there should be 2) which lead to two identical intersections are the tangent lines.
mathman is offline  
September 10th, 2008, 03:07 PM   #5
Senior Member
 
Joined: May 2007

Posts: 402
Thanks: 0

Re: A problem concerning a tangent line and a circle

For a circle of radius $\displaystyle R$ centered at $\displaystyle (x_0, \, y_0)$ to have a tangent of the form $\displaystyle y = k x + l$ you need to have the following condition fulfilled:

$\displaystyle (1+k^2)R^2=(l+k x_0-y_0)^2$

Solving for $\displaystyle k$ yields:

$\displaystyle k_{1,2} = \frac{x_0 (l - y_0)\pm R\sqrt{x_0^2+y_0^2-R^2+l^2-2 l y_0}}{R^2-x_0^2}$

In your problem, you have $\displaystyle l = 2$, $\displaystyle x_0= 2$, $\displaystyle y _0= 1$ and $\displaystyle R = 2$, so the possible values for $\displaystyle k$ are:

$\displaystyle k = \pm \frac{3}{4}$

If you are interested, the tangents touch the circle at points $\displaystyle \left(\frac{16}{5},\, \frac{17}{5}\right)$ and $\displaystyle \left(\frac{16}{5},\, -\frac{7}{5}\right)$.

Last edited by skipjack; February 28th, 2018 at 12:55 PM.
milin is offline  
September 10th, 2008, 03:28 PM   #6
Senior Member
 
Joined: Jul 2008

Posts: 895
Thanks: 0

Re: A problem concerning a tangent line and a circle

The solution may be simpler because of the geometry. The tangent lines intersect at the point (0,1), and are perpendicular to the radius of the given circle. The point(s) of tangency (x,y) and ((0,1) and (5,1) form a right triangle. So, (0,1) and (5,1) form the diameter of an intersecting circle of radius 2.5. center (2.5,1).
Dave is offline  
Reply

  My Math Forum > High School Math Forum > Trigonometry

Tags
circle, line, problem, tangent



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Calculate the center of a circle given a tangent line lauchagonzalez Trigonometry 3 November 19th, 2013 11:40 AM
Tangent Line Problem Shamieh Calculus 6 September 25th, 2013 05:05 PM
Tangent Circle Problem? brian890 Trigonometry 2 April 2nd, 2011 01:31 AM
Line tangent to a circle. Oxymoron Trigonometry 1 July 23rd, 2008 01:13 AM
A tangent line problem derhaus Trigonometry 1 October 15th, 2007 10:38 AM





Copyright © 2019 My Math Forum. All rights reserved.