If z = x + iy, where i = √-1, then what does the equations zz̅ + ∣z ∣ 2 + 4(z + z̅) - 48 = 0 represent?
Circle
Understanding the Complex Number Equation
We are given a complex number $z = x + iy$, where $i = \sqrt{-1}$. The equation we need to analyze is:
\( zz̅ + |z|^2 + 4(z + z̅) - 48 = 0 \)
To determine what this equation represents geometrically in the Cartesian plane (where $x$ and $y$ are the coordinates), we need to express the complex number terms in terms of $x$ and $y$. Let's use the properties of complex numbers:
- The conjugate of \(z = x + iy\) is \(z̅ = x - iy\).
- The product of a complex number and its conjugate is \(zz̅ = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 - (-y^2) = x^2 + y^2\).
- The modulus of a complex number \(z = x + iy\) is \(|z| = \sqrt{x^2 + y^2}\).
- The square of the modulus is \(|z|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2\).
- The sum of a complex number and its conjugate is \(z + z̅ = (x + iy) + (x - iy) = 2x\).
Substituting into the Given Equation
Now, let's substitute these expressions back into the original equation:
\( zz̅ + |z|^2 + 4(z + z̅) - 48 = 0 \)
Substituting the \(x\) and \(y\) terms:
\( (x^2 + y^2) + (x^2 + y^2) + 4(2x) - 48 = 0 \)
Simplifying the Equation
Let's simplify the equation:
\( x^2 + y^2 + x^2 + y^2 + 8x - 48 = 0 \)
Combine like terms:
\( 2x^2 + 2y^2 + 8x - 48 = 0 \)
We can divide the entire equation by 2 to simplify it further:
\( x^2 + y^2 + 4x - 24 = 0 \)
Identifying the Geometric Shape
The simplified equation is \( x^2 + y^2 + 4x - 24 = 0 \). This is an equation in terms of \(x\) and \(y\) in the Cartesian coordinate system.
Let's recall the standard form of equations for common geometric shapes:
- Straight line: \(Ax + By + C = 0\) (linear equation)
- Parabola: Involves one squared term and one linear term (e.g., \(y^2 = 4ax\) or \(x^2 = 4ay\))
- Circle: \( (x - h)^2 + (y - k)^2 = r^2 \) or \( x^2 + y^2 + Dx + Ey + F = 0 \) (coefficients of \(x^2\) and \(y^2\) are equal and positive)
- Pair of straight lines: Often a factorizable quadratic equation
Our simplified equation \( x^2 + y^2 + 4x - 24 = 0 \) has both \(x^2\) and \(y^2\) terms, and their coefficients are equal (both 1). This structure matches the general form of a circle equation \( x^2 + y^2 + Dx + Ey + F = 0 \), where \(D=4\), \(E=0\), and \(F=-24\).
To be even more explicit, we can complete the square to put it in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \):
\( (x^2 + 4x) + y^2 = 24 \)
To complete the square for the \(x\) terms, take half of the coefficient of \(x\) (which is \(4/2 = 2\)) and square it (\(2^2 = 4\)). Add this value to both sides:
\( (x^2 + 4x + 4) + y^2 = 24 + 4 \)
\( (x + 2)^2 + y^2 = 28 \)
This is in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where \(h = -2\), \(k = 0\), and \(r^2 = 28\). This represents a circle with center \((-2, 0)\) and radius \(\sqrt{28}\).
Therefore, the given complex number equation represents a circle.
| Complex Number Term | Equivalent in \(x\) and \(y\) |
|---|---|
| \(z\) | \(x + iy\) |
| \(z̅\) | \(x - iy\) |
| \(zz̅\) | \(x^2 + y^2\) |
| \(|z|^2\) | \(x^2 + y^2\) |
| \(z + z̅\) | \(2x\) |
Conclusion
By substituting \(z = x + iy\) and simplifying the given equation \( zz̅ + |z|^2 + 4(z + z̅) - 48 = 0 \), we obtained the equation \( x^2 + y^2 + 4x - 24 = 0 \). This equation is the general form of a circle.
Revision Table: Identifying Geometric Shapes
| Equation Form | Geometric Shape | Condition |
|---|---|---|
| \(Ax + By + C = 0\) | Straight line | \(A\) or \(B\) is non-zero |
| \(Ax^2 + By^2 + Cx + Dy + E = 0\) | Circle | \(A = B \neq 0\), No \(xy\) term |
| \(Ax^2 + By^2 + Cx + Dy + E = 0\) | Ellipse | \(A \neq B\), \(A\) and \(B\) have same sign, No \(xy\) term |
| \(Ax^2 + By^2 + Cx + Dy + E = 0\) | Hyperbola | \(A\) and \(B\) have opposite signs, No \(xy\) term |
| \(Ax^2 + Cx + Dy + E = 0\) or \(By^2 + Cx + Dy + E = 0\) | Parabola | Only one squared term, No \(xy\) term |
Additional Information: Complex Numbers and Geometry
Complex numbers provide a powerful way to describe geometric objects in a plane. The complex plane, often called the Argand plane, uses the x-axis for the real part (\(x\)) and the y-axis for the imaginary part (\(y\)) of a complex number \(z = x + iy\).
- Points: A complex number \(z\) corresponds to a point \((x, y)\) in the plane.
- Distance: The distance of a point \(z\) from the origin \((0,0)\) is given by its modulus, \(|z| = \sqrt{x^2 + y^2}\).
- Equations: Equations involving complex numbers can often be translated into equations in terms of \(x\) and \(y\), revealing familiar geometric shapes. For example, \(|z - a| = r\) represents a circle centered at the point corresponding to complex number \(a\) with radius \(r\). In our problem, the equation \(|z - (-2)| = \sqrt{28}\) or \(|z + 2| = \sqrt{28}\) is derived from the Cartesian form, showing the center at \((-2, 0)\), which corresponds to the complex number \(-2\).
Analyzing equations like the one given helps understand the relationship between complex algebra and Euclidean geometry.
Important Questions from Algebraic Operations on Complex Numbers
If the point z 1= 1 + i where \({\rm{i}} = \sqrt { - 1} \) is the reflection of a point z 2= x + iy in the line iz̅ - iz = 5, then the point z 2is
z z̅ +(3 - i)z + (3 + i)z̅ + 1 = 0 represents a circle with
What is the number of distinct solutions of the equation z 2+ |z| = 0 (where z is a complex number)?
Which one of the following is a square root of \(\rm 2a+2\sqrt{a^2 + b^2}\) , where a, b ∈ ℝ?
ii = ... will
