Discriminant of a Conic Section
The general equation of a conic section is a second-degree equation in two independent variables (say \(x,y\)) which can be written as
\[f(x,y)=ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. \]
There are several ways of classifying conic sections using the above general equation with the help of the discriminant \(\Delta\) of this equation:
\[\Delta =\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}= abc+2fgh-a{ f }^{ 2 }-b{ g }^{ 2 }-c{ h }^{ 2 },\]
which describes the nature of the conic section. If \(\Delta\) is zero, it represents a degenerate conic section; otherwise, it represents a non-degenerate conic section. This wiki page will give detailed information about the discriminant of a conic section.
Nature of Conic Section
Any second-degree curve equation can be written as
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \qquad (1)\]
or
\[ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, \qquad (2)\]
where \(A,B,C,D,E,F,a,b,c,f,g,h \in \mathbb{R}.\)
Now, let
\[\Delta =\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}= abc+2fgh-a{ f }^{ 2 }-b{ g }^{ 2 }-c{ h }^{ 2 },\]
then the type of conic section that the above equation represents can be found using the discriminant of the equation, which is given by \( B^{2} - 4AC\) for \((1),\) or equivalently, \(h^2–ab\) for \((2).\)
The various conditions regarding the quadratic discriminant are as follows:
If \(\Delta = 0:\)
\( \quad \bullet\) If \(h^2-ab > 0,\) the equation represents two distinct real lines.
\( \quad \bullet\) If \(h^2-ab = 0,\) the equation represents parallel lines.
\( \quad \bullet\) If \(h^2-ab < 0, \) the equation represents non-real lines.
If \(\Delta \neq 0:\)
\( \quad \bullet\) If \(B^2-4AC > 0,\) it represents a hyperbola and a rectangular hyperbola \((A+C=0).\)
\( \quad \bullet\) If \(B^2-4AC = 0, \) the equation represents a parabola.
\( \quad \bullet\) If \(B^2-4AC < 0,\) the equation represents a circle \((A=C, B= 0)\) or an ellipse \((A \neq C).\) \(\big(\)For a real ellipse, \(\frac{\Delta}{a + b} < 0.\big)\)
What type of conic section does the following equation represent: \[5x^2 +10xy+5y^2 +4x+2y + 2 = 0?\]
Equating the given equation with \((1)\) at the top of the page, we have \[A=5, B=10, C=5, D=4, E=2, F=2.\] The discriminant for this equation is \(B^2 – 4AC=10^2-4\cdot 5\cdot 5=0. \qquad \text{(i)}\)
Rewriting the given equation in the form of \((2),\) we have \[5 \cdot x^2+2\cdot 5\cdot xy+5\cdot y^2+2\cdot 2\cdot x+ 2\cdot 1 \cdot y+2=0.\] Equating this with \((2)\) gives \[a=5, b=5, c=2, f=1, g=2, h=5.\] Now, checking \(\Delta,\) we have \[\begin{align} \Delta &=abc+2fgh-a f^2-bg^2-ch^2\\ &=5 \cdot 5 \cdot 2 +2\cdot 1 \cdot 2 \cdot 5- 5\cdot 1^2-5\cdot 2^2-2\cdot 5^2\\ &=50+20-5-20-50=-5 \\&\ne 0. \end{align}\] This confirms \(\Delta\ne 0.\)
Therefore, since \(B^2 – 4AC=0\) from \(\text{(i)},\) the equation represents a parabola. \(_\square\)
The equation \(x^2 + Ky^2 + 8x + 4y + 2 = 0\) represents a circle. What is \(K?\)
Equating the given equation with \((1)\) at the top of the page, we have \[A=1, B=0, C=K, D=8, E=4, F=2.\] Thus, for the equation to represent a circle, the coefficients must simultaneously satisfy the discriminant condition \(B^2 – 4AC< 0\) and also \(A=C.\) The latter condition \(A=C\) implies \(K=1.\) Substituting this into \(B^2 – 4AC< 0\) gives \(0^2-4\cdot 1\cdot 1=-4<0,\) which is true.
Rewriting the given equation in the form of \((2),\) we have \[1 \cdot x^2+2\cdot 0\cdot xy+K\cdot y^2+2\cdot 4\cdot x+ 2\cdot 2y+2=0.\] Equating this with \((2)\) gives \[a=1, b=K, c=2, f=2, g=4, h=0.\] Now, checking \(\Delta,\) we have \[\begin{align} \Delta &=abc+2fgh-a f^2-bg^2-ch^2\\ &=1 \cdot K \cdot 2 + 0-1\cdot 2^2-K\cdot 4^2-0\\ &=-4-14K=-4-14\cdot1 =-18 \\&\ne 0. \end{align}\] This confirms \(\Delta\ne 0.\)
Therefore, \(K=1.\) \(_\square\)
Note: The equation of the circle is \((x+4)^2+(y+2)^2=18.\)
If \(m\) and \(n\) are nonzero real numbers, what is the condition on \(m\) and \(n\) such that the equation \[ \frac{x^2}{m^2}+\frac{y^2}{n^2} = 4\] represents an ellipse that is not a circle?
Equating the given equation with \((1)\) at the top of the page, we have \[A=\frac{1}{m^2}, B=0, C=\frac{1}{n^2}, D=0, E=0, F=-4.\] Thus, for the equation to represent an ellipse that is not a circle, the coefficients must simultaneously satisfy the discriminant condition \(B^2 – 4AC< 0\) and also \(A\ne C.\) The former condition is met because \(B^2 – 4AC=\frac{-4}{m^2 n^2}< 0.\) The latter condition \(A\ne C\) is satisfied if \(m^2\ne n^2.\)
Rewriting the given equation in the form of \((2),\) we have \[\frac{1}{m^2} \cdot x^2+2\cdot 0\cdot xy+\frac{1}{n^2}\cdot y^2+2\cdot 0\cdot x+ 2\cdot 0 \cdot y-4=0.\] Equating this with \((2)\) gives \[a=\frac{1}{m^2}, b=\frac{1}{n^2}, c=-4, f=0, g=0, h=0.\] Now, checking \(\Delta,\) we have \[\begin{align} \Delta &=abc+2fgh-a f^2-bg^2-ch^2\\ &=\frac{1}{m^2} \cdot \frac{1}{n^2} \cdot (-4) + 0-0-0-0\\ &=\frac{-4}{m^2n^2}\\& \ne 0. \end{align}\] This confirms \(\Delta\ne 0.\)
Therefore, the condition for the equation to represent an ellipse that is not a circle is \[m^2 \ne n^2. \ _\square\]
Problem Solving
Check out the following examples based on the discriminant of a conic section.
What type of conic section does the following equation represent: \[3x^2 +10xy+6y^2 +6x+12y + 1 = 0?\]
Equating the given equation with \((1)\) at the top of the page, we have \[A=3, B=10, C=6, D=6, E=12, F=1.\] The discriminant for this equation is \(B^2 – 4AC=10^2-4\cdot 3\cdot 6=28>0. \qquad \text{(i)}\)
Rewriting the given equation in the form of \((2),\) we have \[3 \cdot x^2+2\cdot 5\cdot xy+6\cdot y^2+2\cdot 3\cdot x+ 2\cdot 6 \cdot y+1=0.\] Equating this with \((2)\) gives \[a=3, b=6, c=1, f=6, g=3, h=5.\] Now, checking \(\Delta,\) we have \[\begin{align} \Delta &=abc+2fgh-a f^2-bg^2-ch^2\\ &=3 \cdot 6 \cdot 1 +2\cdot 6 \cdot 3 \cdot 5- 3\cdot 6^2-6\cdot 3^2-1\cdot 5^2\\ &=18+180-108-54-25=11 \\&\ne 0. \end{align}\] This confirms \(\Delta\ne 0.\)
Therefore, since \(B^2 – 4AC>0\) from \(\text{(i)},\) the equation represents a hyperbola. \(_\square\)
Suppose that the equation \(x^2+Jxy+y^2+6x+14y+K = 0\) represents a circle with radius \(2.\) What is \(J+K?\)
Equating the given equation with \((1)\) at the top of the page, we have \[A=1, B=J, C=1, D=6, E=14, F=K.\] Thus, for the equation to represent a circle, the coefficients must simultaneously satisfy the discriminant condition \(B^2 – 4AC< 0\) and also \(A=C\) and \(B=0.\) This last condition \(B=0\) is met if \(J=0.\) The second condition \(A=C\) is already met because we are given \(A=C=1.\) Then the the discriminant condition gives \[B^2 – 4AC=0^2-4\cdot 1\cdot 1=-4< 0,\] which is true.
Thus, rewriting the given equation, we have \[\begin{align} (x+3)^2+(y+7)^2-3^2-7^2+K&=0 \\ (x+3)^2+(y+7)^2&=58-K\\ &=r^2, \end{align}\] where \(r=2\) is the radius of the circle. Hence, \(K=58-2^2=54,\) implying \[J+K=0+54=54. \ _\square\]
Try the following problems.
What does the equation \(xy=1\) represent?
\[ 6x^2-\alpha xy-3y^2-24x+3y+\beta=0 \]
The equation above represents a pair of lines that intersect on the \(x\)-axis. Find the value of \(20\alpha-\beta\).
There exist two ordered triples \((a,b,c)\) for which \(4x^2-4xy+ay^2+bx+cy+1=0\) represents a pair of identical straight lines in the \(xy\)-plane.
If these triples are \((a_1,b_1,c_1)\) and \((a_2,b_2,c_2)\), then find the value of \(a_1+b_1+c_1+a_2+b_2+c_2\).
\[\begin{align} f(x , y) &= nx^2 + \left(\dfrac{nm}{2}\right)xy + my^2 + 4x + 2y + 1 = 0\\ g(x , y) &= nx^2 + (n + m)xy + my^2 + 4x + 2y + 1 = 0\\ h(x , y) &= nx^2 + (n - m)xy + my^2 + 4x + 2y + 1 = 0 \end{align}\]
Let the graphs \(f, g, h\) be a parabola, a hyperbola, and an ellipse, respectively.
If \(n\) and \(m\) are positive integers satisfying the constraints above, compute \(2n + m\).