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發問:
let z be a complex number the standard form of circle is l z-(a+bi) l=r how about straight line,hyperbola,parabola and ellipse? thx!
最佳解答:
For straight line, we may make reference to its vector equation which is: r = a + kb where a and b are fixed vectors with the latter representing the orientation of the line and k is a variable scalar. Then, applying this concept onto complex plane, we can have the equation in the form: z = z1 + kz2 where z1 and z2 are fixed complex numbers with z2 representing the direction of the line. Then, for the rest, we have to first describe how a point runs so that it can move out those loci: Parabola: The perpendicular distance from a straight line (known as the directrix) is always equal to that from a fixed point (known as the focus). For convenience, we describe only horizontal parabola and vertical parabola: Horizontal (Directrix is vertical): Re(z) - K = |z - z1| with K being a real constant and z1 representing the focus on the Argand diagram. Vertical (Directrix is horizontal): Im(z) - K = |z - z1| with K being a real constant and z1 representing the focus on the Argand diagram. Ellipse: The total distance of the point from two fixed points (known as foci) is a constant. So the equation is: |z - z1| + |z - z2| = K where z1 and z2 are complex nos. representing the foci in the Argand diagram and K is a real constant. Hyperbola: The difference between the distances of the point from two fixed points (known as foci) is a constant. So the equation is: |z - z1| - |z - z2| = K where z1 and z2 are complex nos. representing the foci in the Argand diagram and K is a real constant.
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complex number problem發問:
let z be a complex number the standard form of circle is l z-(a+bi) l=r how about straight line,hyperbola,parabola and ellipse? thx!
最佳解答:
For straight line, we may make reference to its vector equation which is: r = a + kb where a and b are fixed vectors with the latter representing the orientation of the line and k is a variable scalar. Then, applying this concept onto complex plane, we can have the equation in the form: z = z1 + kz2 where z1 and z2 are fixed complex numbers with z2 representing the direction of the line. Then, for the rest, we have to first describe how a point runs so that it can move out those loci: Parabola: The perpendicular distance from a straight line (known as the directrix) is always equal to that from a fixed point (known as the focus). For convenience, we describe only horizontal parabola and vertical parabola: Horizontal (Directrix is vertical): Re(z) - K = |z - z1| with K being a real constant and z1 representing the focus on the Argand diagram. Vertical (Directrix is horizontal): Im(z) - K = |z - z1| with K being a real constant and z1 representing the focus on the Argand diagram. Ellipse: The total distance of the point from two fixed points (known as foci) is a constant. So the equation is: |z - z1| + |z - z2| = K where z1 and z2 are complex nos. representing the foci in the Argand diagram and K is a real constant. Hyperbola: The difference between the distances of the point from two fixed points (known as foci) is a constant. So the equation is: |z - z1| - |z - z2| = K where z1 and z2 are complex nos. representing the foci in the Argand diagram and K is a real constant.
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