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Math Help - Coordinates of the midpoint of intersecting lines

  1. #1
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    Coordinates of the midpoint of intersecting lines

    Find the coordinates of the mid-point of the straight line joining the points of intersection of the curve x^2+2y^2+5x=68 and the line 2y+3x=9
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  2. #2
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    x^2 + 2y^2 + 5x = 68 and y = \frac{9-3x}{2}.

    Substituting the second into the first gives

    x^2 + \left(\frac{9-3x}{2}\right)^2 + 5x = 68

    \frac{4x^2}{4} + \frac{81 -54x + 9x^2}{4} + \frac{20x}{4} = \frac{272}{4}

    \frac{13x^2 - 34x - 191}{4} = 0

    13x^2 - 34x - 191 = 0


    Now solve for x and substitute back to find the corresponding y values. This will give you the two points, so you can use the midpoint rule to find the midpoint.
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  3. #3
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    The kind of obvious thing to do is to find the two point of intersection. From 2y+ 3x= 9, y= 9/2- (3/2)x. Replace y in x^2+ 2y^2+ 5x= 68 and you get a quadratic equation for x which should have two solutions. Once you have found those two points of intersection, find the point midway between them.

    Do you know that the point midway between (x_0, y_0) and (x_1, y_1) is \left(\frac{x_0+ x_1}{2}, \frac{y_0+ y_1}{2}\right)?
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  4. #4
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    Sorry guys, I misunderstood that the question stated there was only 1 point of intersection...

    didnt notice that the word was 'points'.
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