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Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:-

$$ax + by + c = 0 \tag 1$$

$$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$

What are the conditions that can be imposed on the coefficients such that the solution of x (or y) has exactly one root (not equal or double repeated roots).

The following is an example to illustrate my point:-

$$x + y = 6 \tag 3$$

$$x^2 - y^2 = 12 \tag 4$$

At one glance of the above, we are expecting two roots (because one equation is quadratic) but it turns out that we can get $(x, y) = (4, 2)$ only.

Michael Hardy
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Mick
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  • Here's an oddity of MathJax. You normally use a hyphen to cause a minus sign to appear, as in $x^2-y^2$ (which obviously does not look like a hyphen, as in $x^2$-$y^2$), but if you use an en-dash or an actual minus sign within MathJax instead of a hyphen, then you see this: $x^2 – y^2$, which doesn't have the spacing that you expect with a minus sign. (Hence my edit to this answer.) $\qquad$ – Michael Hardy Mar 27 '16 at 04:18
  • You only get one solution for the simultaneous system because the line $y=6-x$ cuts the upper right side of the curve $\sqrt{x^{2}-12}$ - See:https://www.desmos.com/calculator/7roo1lvmag – NoChance Mar 27 '16 at 04:24
  • @MichaelHardy Got it. – Mick Mar 27 '16 at 04:26
  • @NoChance Good tool in explaining the particular case. How about the general situation? – Mick Mar 27 '16 at 04:31
  • Maybe you can check on "Nonlinear Simultaneous Equations", for example:http://www.purplemath.com/modules/syseqgen3.htm – NoChance Mar 27 '16 at 04:42

1 Answers1

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$a x + by + c = 0$ is a line.

$A x^2 + B xy + C y^2 + Dx + Ey +F = 0$ Is some sort of conic section, that has been rotated off of the standard.

If $B^2 - 4AC = 0$ you have a parabola.

I line will intersect a parabola it two places, unless the line is tangent to the parabola (one place), is parallel to the axis of symmetry (one place), or fails to intersect (no solution).

If $B^2 - 4AC < 0$ you have an ellipse.

A line will intersect an ellipse in two places, unless it is tangent, or it fails to intersect.

If $B^2 - 4AC >0$ you have an hyperbola.

A line will intersect a hyperbola in two places, unless the line is parallel to an asymptote, or is an asymptote.

In the example, the quadratic is a hyperbola, and the line is parallel to its asymptote.

user317176
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  • I see your point. That means we have to draw the graphs and see how they intersect before we can tell the number of roots. In other words, there is no way of predicting the exact number of roots by just looking at the coefficients. – Mick Mar 28 '16 at 04:26
  • One more thing, you have used all the three possible cases of the discriminant to classify the conics. What happens if the general quadratic equation represents a pair of straight lines? – Mick Mar 28 '16 at 04:27