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Math Help - Challenging Question - help a brother

  1. #1
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    Challenging Question - help a brother

    Hello,
    thanks in advance for anyone who can help me.

    this question relates to linear quadratic systems

    the question is:


    Generalize your results for a circle of radius r and the line y = x+k


    I thought maybe to help, I would put the question before this one, which relates to this question.
    It goes like this

    consider the circle x^2 + y^2 = 25 and the line y = x +k, where k is any real number. Determine the values of k for which the line will intersect the circle in one, two, or no points. Repeat for the circle x^2 + y^2 = 49 and the line y = x+k.

    I figured out how to do this by subbing, x+k into the other equation, and then using discriminants to find out the values of k that would work.

    im confused on the generalization question though.

    Again, thanks to anyone who can help me
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  2. #2
    MHF Contributor

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    You really should graph this problem.
    The line  y = x + k is tangent to x^2 + y^2 = {25}
    at \left( { - \sqrt {12.5} ,\sqrt {12.5} } \right)\quad \& \quad \left( {\sqrt {12.5} , - \sqrt {12.5} } \right).

    If you use those points to find values of k and if you graph this correctly, you will see the answer.
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  3. #3
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    Quote Originally Posted by thejoester View Post
    ...

    Generalize your results for a circle of radius r and the line y = x+k

    ...
    Hello,

    the circle has the equation:
    x^2+y^2=r^2 and the line has the equation: y=x+k.

    To calculate the intercepts you plug in the term of the line into the equation of the xircle:
    x^2+(x+k)^2=r^2 . After a few steps you'll get:

    2x^2+2kx+k^2-r^2=0. Solve for x. Use the formula to solve quadratic equations:

    x=\frac{-2k \pm \sqrt{4k^2-4 \cdot 2 \cdot (k^2-r^2)}}{2 \cdot  2}= \frac{-2k \pm \sqrt{8r^2-4k^2}}{4}= \frac{-k \pm \sqrt{2r^2-k^2}}{2}

    You get 2 intercepts if the discriminant is greater than zero: {2r^2-k^2}>0
    You get 1 intercept if the discriminant equals zero: {2r^2-k^2}=0
    You get no intercepts if the discriminant is negative: {2r^2-k^2}<0

    Solve for k to answer the question.

    EB
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