Equation of a line tangent to a circle

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Discussion Overview

The discussion revolves around finding the value of \( m \) such that the circle defined by the equation \( x^2 + y^2 - 4x + 2y + m = 0 \) is tangent to the line \( y = x + 1 \). Participants explore different methods to approach the problem, including geometric interpretations and algebraic manipulations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant expresses confusion about substituting \( y = x + 1 \) into the circle's equation to find \( m \).
  • Another participant suggests using the quadratic formula to analyze the discriminant \( b^2 - 4ac \) to determine the nature of the solutions, indicating that a zero discriminant would imply tangency.
  • A further response proposes a geometric approach, suggesting that the center of the circle can be found by completing the square, leading to the identification of the center and the relationship between the radius and the tangent line.
  • This participant also outlines steps to find the tangent point and calculate the radius, linking it back to the value of \( m \).

Areas of Agreement / Disagreement

There is no consensus on a single method to solve for \( m \). Participants present different approaches, and while some methods are discussed in detail, the discussion remains unresolved regarding the best or most straightforward solution.

Contextual Notes

Participants have not fully resolved the assumptions regarding the relationship between the circle's radius and the tangent line, nor have they clarified the implications of the quadratic discriminant in this specific context.

Armela
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The circle x^2 +y^2 -4x+2y+m=0 is tangent with the line y=x+1.Find m.

p.s : I know that o should solve it from the equations of two lines but i really get confused when i substitute the y :/ .
Thanx :)
 
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Hi Armela,

Welcome to MHB! :)

Geometry is not my strong suit at all, but I did find http://planetmath.org/EquationOfTangentOfCircle.html for you which might be useful.

Jameson
 
Armela said:
The circle x^2 +y^2 -4x+2y+m=0 is tangent with the line y=x+1.Find m.

p.s : I know that o should solve it from the equations of two lines but i really get confused when i substitute the y :/ .
Thanx :)

When you substitute $y=x+1$ in the equation of the circle, you get a quadratic equation for $x$. Solve that quadratic equation using the "$\sqrt{b^2-4ac}$" formula (the solution will involve the constant $m$). If $b^2-4ac$ is positive then there are two solutions to the equation, meaning that the line cuts the circle in two points. If it is negative then there are no solutions, meaning that the line misses the circle. But if it is zero then there is just one (repeated) solution, meaning that the line is tangent to the circle.
 
Armela said:
The circle x^2 +y^2 -4x+2y+m=0 is tangent with the line y=x+1.Find m.

p.s : I know that o should solve it from the equations of two lines but i really get confused when i substitute the y :/ .
Thanx :)


Maybe I understand your remark about the two lines completely wrong, but here comes a way to use actually two lines:

1. Determine the center of the circle by completing the squares. You should come out with:

$\displaystyle{(x-2)^2+(y+1)^2= 5-m}$

So the center is at C(2, -1)

2. If the given line is a tangent to the circle then the radius of the circle is perpendicular to the given line at the tangent point T.
The given line has the slope m = 1 therefore the line frome the center C to the tangent point T has the slope m = -1.
Determine the equation of the line CT. You should come out with

$y = -x+1$

3. Determine the intercept between the given line and CT to get the coordinates of T. You should come out with $T(0, 1)$.

4. Calculate the distance $r=|\overline{CT}|$.

Since $r^2=5-m$ you are able to determine the value of m.
 

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