Conic Sections formulae using Complex Numbers
$begingroup$
I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in Coordinate Geometry (Conic Sections). These include the following:
Equation of lines in different forms (parametric/non-parametric)$$$$
Equations of Circles with various conditions (for example centered at $z_0$, or orthogonal to another circle and so on)$$$$
Equations of Ellipses and Hyperbolas and so on.
$$$$
There are several other applications/equations mentioned which I haven't written here. The trouble I'm facing is that the book I use for Complex Numbers (Algebra for JEE Main and Advanced, by SK Goyal) only has the formulae listed without showing the derivations. I find it extremely hard to just accept and use a formula/result without knowing how it came into existence.
$$$$
I would be grateful if somebody could please mention a source/reference book from which I could actually learn how to derive/reach all the formulae used for represinting Conics using Complex Numbers. The book does not necessarily have to match the level of the JEE Advanced Examination; it can be higher than that too. However I would prefer it if the book was at the level suitable for the JEE Advanced level only.
$$$$Many thanks in advance!
$$$$
Edit:$$$$ The results mentioned in my book are as follows:
$$$$
$1) $The equation of the line joining $z_1$ and $z_2$ is $$z(bar{z_1}-bar{z_2})-bar{z}(z_1-z_2)+ z_1bar z_2-z_2bar z_1=0 text{ (non parametric form).}$$
$$$$
$2) $Three points are collinear if $$begin{vmatrix}z_1&bar{z_1}&1\z_2&bar {z_2}& 1\z_3&bar {z_3}& 1end{vmatrix}=0$$
$$$$
$3)$$bar{a}z+bar z a+b=0$ where $bin mathbb R$ describes the equation of a straight line (I don't know what $a$ is, nor what $iota b$ is).
$$$$
$4)$ The real and complex slope (I don't know what is meant by 'real' and 'complex') of the line $bar{a}z+bar z a+b=0$ are $-dfrac{Re(a)}{Im(a)}$ and $-dfrac{a}{bar a}$ where $bin mathbb R$.
$$$$
$5) $If the lines $bar{a}z+bar z a+k_1=0$ and $bar{b}z+bar b a+k_2=0$ $k_1,k_2in mathbb R$ are perpendicular to each other, then $$bar{a}b+bar ba=0$$
$$$$
$6)$ $zbar z +abar z +bar az+k=0 $ where $kin mathbb R$ represents a circle with center $-a$ and radius $sqrt{|a|^2-k}$.
$$$$
$7)$ If $|z-z_1|+|z-z_2|=2a$ where $2a>|z_1-z_2|$ then $ z $ represents an ellipse with foci at $z_1 text{ and }z_2$ and $ain mathbb R^+ $.
$$$$
$8)$ If $|z-z_1|-|z-z_2|=2a$ where $2a<|z_1-z_2|$ then $z$ represents a hyperbola with foci at $z_1$ and $z_2$.
$$$$
$9)$ Equation of all circles orthogonal to $|z-z_1|=r_1text{ and }|z-z_2|=r_2$ is (nothing further is mentioned).
$$$$
$10)$ $left |dfrac{z-z_1}{z-z_2}right | = k$ is a circle if $kneq 1$ and will represent a line if $k=1$.
$$$$
$11)$ The equation $|z-z_1|^2+|z-z_2|^2=k$ will represent a circle if $kgeq frac12 |z_1-z_2|^2$.
$$$$
$12)$ If $argleft(dfrac{(z_2-z_3)(z_1-z_4)}{(z_1-z_3)(z_2-z_4)}right)=0, pm pi$, then the points $z_1,z_2,z_3,z_4$ are concyclic.
algebra-precalculus complex-numbers soft-question conic-sections
asked Oct 7 '16 at 11:30
IshanIshan
283
$endgroup$
|
show 2 more comments
$begingroup$
I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in Coordinate Geometry (Conic Sections). These include the following:
Equation of lines in different forms (parametric/non-parametric)$$$$
Equations of Circles with various conditions (for example centered at $z_0$, or orthogonal to another circle and so on)$$$$
Equations of Ellipses and Hyperbolas and so on.
$$$$
There are several other applications/equations mentioned which I haven't written here. The trouble I'm facing is that the book I use for Complex Numbers (Algebra for JEE Main and Advanced, by SK Goyal) only has the formulae listed without showing the derivations. I find it extremely hard to just accept and use a formula/result without knowing how it came into existence.
$$$$
I would be grateful if somebody could please mention a source/reference book from which I could actually learn how to derive/reach all the formulae used for represinting Conics using Complex Numbers. The book does not necessarily have to match the level of the JEE Advanced Examination; it can be higher than that too. However I would prefer it if the book was at the level suitable for the JEE Advanced level only.
$$$$Many thanks in advance!
$$$$
Edit:$$$$ The results mentioned in my book are as follows:
$$$$
$1) $The equation of the line joining $z_1$ and $z_2$ is $$z(bar{z_1}-bar{z_2})-bar{z}(z_1-z_2)+ z_1bar z_2-z_2bar z_1=0 text{ (non parametric form).}$$
$$$$
$2) $Three points are collinear if $$begin{vmatrix}z_1&bar{z_1}&1\z_2&bar {z_2}& 1\z_3&bar {z_3}& 1end{vmatrix}=0$$
$$$$
$3)$$bar{a}z+bar z a+b=0$ where $bin mathbb R$ describes the equation of a straight line (I don't know what $a$ is, nor what $iota b$ is).
$$$$
$4)$ The real and complex slope (I don't know what is meant by 'real' and 'complex') of the line $bar{a}z+bar z a+b=0$ are $-dfrac{Re(a)}{Im(a)}$ and $-dfrac{a}{bar a}$ where $bin mathbb R$.
$$$$
$5) $If the lines $bar{a}z+bar z a+k_1=0$ and $bar{b}z+bar b a+k_2=0$ $k_1,k_2in mathbb R$ are perpendicular to each other, then $$bar{a}b+bar ba=0$$
$$$$
$6)$ $zbar z +abar z +bar az+k=0 $ where $kin mathbb R$ represents a circle with center $-a$ and radius $sqrt{|a|^2-k}$.
$$$$
$7)$ If $|z-z_1|+|z-z_2|=2a$ where $2a>|z_1-z_2|$ then $ z $ represents an ellipse with foci at $z_1 text{ and }z_2$ and $ain mathbb R^+ $.
$$$$
$8)$ If $|z-z_1|-|z-z_2|=2a$ where $2a<|z_1-z_2|$ then $z$ represents a hyperbola with foci at $z_1$ and $z_2$.
$$$$
$9)$ Equation of all circles orthogonal to $|z-z_1|=r_1text{ and }|z-z_2|=r_2$ is (nothing further is mentioned).
$$$$
$10)$ $left |dfrac{z-z_1}{z-z_2}right | = k$ is a circle if $kneq 1$ and will represent a line if $k=1$.
$$$$
$11)$ The equation $|z-z_1|^2+|z-z_2|^2=k$ will represent a circle if $kgeq frac12 |z_1-z_2|^2$.
$$$$
$12)$ If $argleft(dfrac{(z_2-z_3)(z_1-z_4)}{(z_1-z_3)(z_2-z_4)}right)=0, pm pi$, then the points $z_1,z_2,z_3,z_4$ are concyclic.
algebra-precalculus complex-numbers soft-question conic-sections
asked Oct 7 '16 at 11:30
IshanIshan
283
$endgroup$
$begingroup$
It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:04
$begingroup$
You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:07
$begingroup$
@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
$endgroup$
– Ishan
Oct 7 '16 at 14:01
$begingroup$
@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
$endgroup$
– Ishan
Oct 7 '16 at 14:07
$begingroup$
Have you looked at any of the links I gave? Were any of them helpful?
$endgroup$
– Gerry Myerson
Oct 7 '16 at 21:24
|
show 2 more comments
$begingroup$
I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in Coordinate Geometry (Conic Sections). These include the following:
Equation of lines in different forms (parametric/non-parametric)$$$$
Equations of Circles with various conditions (for example centered at $z_0$, or orthogonal to another circle and so on)$$$$
Equations of Ellipses and Hyperbolas and so on.
$$$$
There are several other applications/equations mentioned which I haven't written here. The trouble I'm facing is that the book I use for Complex Numbers (Algebra for JEE Main and Advanced, by SK Goyal) only has the formulae listed without showing the derivations. I find it extremely hard to just accept and use a formula/result without knowing how it came into existence.
$$$$
I would be grateful if somebody could please mention a source/reference book from which I could actually learn how to derive/reach all the formulae used for represinting Conics using Complex Numbers. The book does not necessarily have to match the level of the JEE Advanced Examination; it can be higher than that too. However I would prefer it if the book was at the level suitable for the JEE Advanced level only.
$$$$Many thanks in advance!
$$$$
Edit:$$$$ The results mentioned in my book are as follows:
$$$$
$1) $The equation of the line joining $z_1$ and $z_2$ is $$z(bar{z_1}-bar{z_2})-bar{z}(z_1-z_2)+ z_1bar z_2-z_2bar z_1=0 text{ (non parametric form).}$$
$$$$
$2) $Three points are collinear if $$begin{vmatrix}z_1&bar{z_1}&1\z_2&bar {z_2}& 1\z_3&bar {z_3}& 1end{vmatrix}=0$$
$$$$
$3)$$bar{a}z+bar z a+b=0$ where $bin mathbb R$ describes the equation of a straight line (I don't know what $a$ is, nor what $iota b$ is).
$$$$
$4)$ The real and complex slope (I don't know what is meant by 'real' and 'complex') of the line $bar{a}z+bar z a+b=0$ are $-dfrac{Re(a)}{Im(a)}$ and $-dfrac{a}{bar a}$ where $bin mathbb R$.
$$$$
$5) $If the lines $bar{a}z+bar z a+k_1=0$ and $bar{b}z+bar b a+k_2=0$ $k_1,k_2in mathbb R$ are perpendicular to each other, then $$bar{a}b+bar ba=0$$
$$$$
$6)$ $zbar z +abar z +bar az+k=0 $ where $kin mathbb R$ represents a circle with center $-a$ and radius $sqrt{|a|^2-k}$.
$$$$
$7)$ If $|z-z_1|+|z-z_2|=2a$ where $2a>|z_1-z_2|$ then $ z $ represents an ellipse with foci at $z_1 text{ and }z_2$ and $ain mathbb R^+ $.
$$$$
$8)$ If $|z-z_1|-|z-z_2|=2a$ where $2a<|z_1-z_2|$ then $z$ represents a hyperbola with foci at $z_1$ and $z_2$.
$$$$
$9)$ Equation of all circles orthogonal to $|z-z_1|=r_1text{ and }|z-z_2|=r_2$ is (nothing further is mentioned).
$$$$
$10)$ $left |dfrac{z-z_1}{z-z_2}right | = k$ is a circle if $kneq 1$ and will represent a line if $k=1$.
$$$$
$11)$ The equation $|z-z_1|^2+|z-z_2|^2=k$ will represent a circle if $kgeq frac12 |z_1-z_2|^2$.
$$$$
$12)$ If $argleft(dfrac{(z_2-z_3)(z_1-z_4)}{(z_1-z_3)(z_2-z_4)}right)=0, pm pi$, then the points $z_1,z_2,z_3,z_4$ are concyclic.
algebra-precalculus complex-numbers soft-question conic-sections
asked Oct 7 '16 at 11:30
IshanIshan
283
$endgroup$
I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in Coordinate Geometry (Conic Sections). These include the following:
Equation of lines in different forms (parametric/non-parametric)$$$$
Equations of Circles with various conditions (for example centered at $z_0$, or orthogonal to another circle and so on)$$$$
Equations of Ellipses and Hyperbolas and so on.
$$$$
There are several other applications/equations mentioned which I haven't written here. The trouble I'm facing is that the book I use for Complex Numbers (Algebra for JEE Main and Advanced, by SK Goyal) only has the formulae listed without showing the derivations. I find it extremely hard to just accept and use a formula/result without knowing how it came into existence.
$$$$
I would be grateful if somebody could please mention a source/reference book from which I could actually learn how to derive/reach all the formulae used for represinting Conics using Complex Numbers. The book does not necessarily have to match the level of the JEE Advanced Examination; it can be higher than that too. However I would prefer it if the book was at the level suitable for the JEE Advanced level only.
$$$$Many thanks in advance!
$$$$
Edit:$$$$ The results mentioned in my book are as follows:
$$$$
$1) $The equation of the line joining $z_1$ and $z_2$ is $$z(bar{z_1}-bar{z_2})-bar{z}(z_1-z_2)+ z_1bar z_2-z_2bar z_1=0 text{ (non parametric form).}$$
$$$$
$2) $Three points are collinear if $$begin{vmatrix}z_1&bar{z_1}&1\z_2&bar {z_2}& 1\z_3&bar {z_3}& 1end{vmatrix}=0$$
$$$$
$3)$$bar{a}z+bar z a+b=0$ where $bin mathbb R$ describes the equation of a straight line (I don't know what $a$ is, nor what $iota b$ is).
$$$$
$4)$ The real and complex slope (I don't know what is meant by 'real' and 'complex') of the line $bar{a}z+bar z a+b=0$ are $-dfrac{Re(a)}{Im(a)}$ and $-dfrac{a}{bar a}$ where $bin mathbb R$.
$$$$
$5) $If the lines $bar{a}z+bar z a+k_1=0$ and $bar{b}z+bar b a+k_2=0$ $k_1,k_2in mathbb R$ are perpendicular to each other, then $$bar{a}b+bar ba=0$$
$$$$
$6)$ $zbar z +abar z +bar az+k=0 $ where $kin mathbb R$ represents a circle with center $-a$ and radius $sqrt{|a|^2-k}$.
$$$$
$7)$ If $|z-z_1|+|z-z_2|=2a$ where $2a>|z_1-z_2|$ then $ z $ represents an ellipse with foci at $z_1 text{ and }z_2$ and $ain mathbb R^+ $.
$$$$
$8)$ If $|z-z_1|-|z-z_2|=2a$ where $2a<|z_1-z_2|$ then $z$ represents a hyperbola with foci at $z_1$ and $z_2$.
$$$$
$9)$ Equation of all circles orthogonal to $|z-z_1|=r_1text{ and }|z-z_2|=r_2$ is (nothing further is mentioned).
$$$$
$10)$ $left |dfrac{z-z_1}{z-z_2}right | = k$ is a circle if $kneq 1$ and will represent a line if $k=1$.
$$$$
$11)$ The equation $|z-z_1|^2+|z-z_2|^2=k$ will represent a circle if $kgeq frac12 |z_1-z_2|^2$.
$$$$
$12)$ If $argleft(dfrac{(z_2-z_3)(z_1-z_4)}{(z_1-z_3)(z_2-z_4)}right)=0, pm pi$, then the points $z_1,z_2,z_3,z_4$ are concyclic.
algebra-precalculus complex-numbers soft-question conic-sections
algebra-precalculus complex-numbers soft-question conic-sections
asked Oct 7 '16 at 11:30
IshanIshan
283
asked Oct 7 '16 at 11:30
IshanIshan
283
edited Oct 7 '16 at 17:09
Ishan
asked Oct 7 '16 at 11:30
IshanIshan
283
asked Oct 7 '16 at 11:30
IshanIshan
283
asked Oct 7 '16 at 11:30
IshanIshan
283
283
$begingroup$
It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:04
$begingroup$
You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:07
$begingroup$
@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
$endgroup$
– Ishan
Oct 7 '16 at 14:01
$begingroup$
@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
$endgroup$
– Ishan
Oct 7 '16 at 14:07
$begingroup$
Have you looked at any of the links I gave? Were any of them helpful?
$endgroup$
– Gerry Myerson
Oct 7 '16 at 21:24
|
show 2 more comments
$begingroup$
It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:04
$begingroup$
You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:07
$begingroup$
@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
$endgroup$
– Ishan
Oct 7 '16 at 14:01
$begingroup$
@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
$endgroup$
– Ishan
Oct 7 '16 at 14:07
$begingroup$
Have you looked at any of the links I gave? Were any of them helpful?
$endgroup$
– Gerry Myerson
Oct 7 '16 at 21:24
$begingroup$
It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:04
$begingroup$
It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:04
$begingroup$
You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:07
$begingroup$
You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
$endgroup$
– Gerry Myerson
Oct 7 '16 at 12:07
$begingroup$
@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
$endgroup$
– Ishan
Oct 7 '16 at 14:01
$begingroup$
@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
$endgroup$
– Ishan
Oct 7 '16 at 14:01
$begingroup$
@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
$endgroup$
– Ishan
Oct 7 '16 at 14:07
$begingroup$
@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
$endgroup$
– Ishan
Oct 7 '16 at 14:07
$begingroup$
Have you looked at any of the links I gave? Were any of them helpful?
$endgroup$
– Gerry Myerson
Oct 7 '16 at 21:24
$begingroup$
Have you looked at any of the links I gave? Were any of them helpful?
$endgroup$
– Gerry Myerson
Oct 7 '16 at 21:24
|
show 2 more comments
1 Answer
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$begingroup$
Try R D sharma.They combine pretty much every chapters.Also try any previous year question paper if you are are preparing for JEE Advanced.I am repeating this year and last year i found out that the JEE mains and advanced are pretty differentI hope you passed your exams.
answered Apr 14 '18 at 9:55
xoxoxoxo
1
$endgroup$
add a comment |
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$begingroup$
Try R D sharma.They combine pretty much every chapters.Also try any previous year question paper if you are are preparing for JEE Advanced.I am repeating this year and last year i found out that the JEE mains and advanced are pretty differentI hope you passed your exams.
answered Apr 14 '18 at 9:55
xoxoxoxo
1
$endgroup$
add a comment |
$begingroup$
Try R D sharma.They combine pretty much every chapters.Also try any previous year question paper if you are are preparing for JEE Advanced.I am repeating this year and last year i found out that the JEE mains and advanced are pretty differentI hope you passed your exams.
answered Apr 14 '18 at 9:55
xoxoxoxo
1
$endgroup$
add a comment |
$begingroup$
Try R D sharma.They combine pretty much every chapters.Also try any previous year question paper if you are are preparing for JEE Advanced.I am repeating this year and last year i found out that the JEE mains and advanced are pretty differentI hope you passed your exams.
answered Apr 14 '18 at 9:55
xoxoxoxo
1
$endgroup$
Try R D sharma.They combine pretty much every chapters.Also try any previous year question paper if you are are preparing for JEE Advanced.I am repeating this year and last year i found out that the JEE mains and advanced are pretty differentI hope you passed your exams.
answered Apr 14 '18 at 9:55
xoxoxoxo
1
answered Apr 14 '18 at 9:55
xoxoxoxo
1
answered Apr 14 '18 at 9:55
xoxoxoxo
1
answered Apr 14 '18 at 9:55
xoxoxoxo
1
1
add a comment |
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It might help if you would write out a few of the formulas, so we would know what sort of reference to look for.
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– Gerry Myerson
Oct 7 '16 at 12:04
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You could also see whether math.stackexchange.com/questions/481582/… is any help, or math.stackexchange.com/questions/786215/…, or people.eecs.berkeley.edu/~wkahan/Math185/Conics.pdf, or anything else you get by typing Conic Sections formulae using Complex Numbers into the internet.
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– Gerry Myerson
Oct 7 '16 at 12:07
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@GerryMyerson Thanks for responding Sir. I've edited my question to include the results mentioned in the book. These were actually mentioned as extra points, but these are frequently used to solved questions in the JEE papers. I would be truly grateful if you could please show me how to derive all these results, or give me a source from which I could learn how to derive them. Once again, many thanks Sir!
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– Ishan
Oct 7 '16 at 14:01
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@GerryMyerson PS. Sir I've tried multiple times to format the edit properly, but it just isn't working. I hope you will excuse it Sir.
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– Ishan
Oct 7 '16 at 14:07
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Have you looked at any of the links I gave? Were any of them helpful?
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– Gerry Myerson
Oct 7 '16 at 21:24