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Math Help - Quadratics Question

  1. #1
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    Quadratics Question

    I'm having some trouble with this question.


    1. a) The line y = x -1 intersects the circle x^2 + y^2 = 25 at two points. This line is called a secant. Find the coordinates of the 2 points of intersection.

    b) For what value(s) of k will the line y = x + k be a tangent to the circle x^2 + y^2 = 25?
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  2. #2
    Member Jonboy's Avatar
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    for the first one, you could just use basic substitution.

    You know y = x - 1, so fill x - 1 for y in the second equation:

    x^2 + (x - 1)^2 = 25

    Foil the (x - 1)^2: x^2 + (x^2 -2x + 1) = 25

    Simplify the LHS: 2x^2 -2x + 1 = 25

    Get a quadratic equation: 2x^2 -2x -24 = 0

    Divide both sides by 2: x^2 - x - 12 = 0

    Factor it, getting: (x - 4)(x + 3) = 0

    Solve for x, which obviously are the two x values of the two coordinates.

    When you get the two x values, fill it in y = x - 1 to get the two y coordinates.

    There ya go!
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  3. #3
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    I also need help on the second one.


    Can anyone help?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by finalfantasy View Post
    b) For what value(s) of k will the line y = x + k be a tangent to the circle x^2 + y^2 = 25?
    again, as Jonboy said, substitute (x + k) for y in the equation of the circle, this will give you the points of intersection.

    we get: x^2 + (x + k)^2 = 25

    \Rightarrow x^2 + x^2 + 2kx + k^2 - 25 = 0

    \Rightarrow 2x^2 + 2kx + \left( k^2 - 25 \right) = 0

    now, if the line is tangent to the circle, it means it touches the circle at only one point. but that happens only if our quadratic has one solution. and in order for that to happen, we must have the discriminant be zero (can you see why? do you know what the discriminant is?)

    thus we want: 4k^2 - 8 \left( k^2 - 25 \right) = 0

    now solve for k
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  5. #5
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    Is the answer 5 and - 5 .. and 0?

    Thanks!
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by finalfantasy View Post
    Is the answer 5 and - 5 .. and 0?

    Thanks!
    no. by the way, be specific as to which question you're offering the answer for
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