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Intersection of tangents

  • #1
utkarshakash
Gold Member
855
13

Homework Statement


Two points represented by [itex]z_{1}, z_{2}[/itex] lie on circle |z|=1, the tangents to the circle at these points meet at the point represented by

Homework Equations





The Attempt at a Solution


Let the tangents meet at the point [itex]z_{3}[/itex]. The centre of the given circle is (0,0).
∴ [itex]z_{1}, z_{2}, z_{3} and 0[/itex] are concyclic. Now what to do next?
 

Answers and Replies

  • #2
ehild
Homework Helper
15,477
1,854

Homework Statement


Two points represented by [itex]z_{1}, z_{2}[/itex] lie on circle |z|=1, the tangents to the circle at these points meet at the point represented by

Homework Equations





The Attempt at a Solution


Let the tangents meet at the point [itex]z_{3}[/itex]. The centre of the given circle is (0,0).
[itex]z_{1}, z_{2}, z_{3} and 0[/itex] are concyclic. Now what to do next?
It is not true. z1 and z2 are on the circle which centre is (0,0) z3 can not lie on the circle.

Find the equations of the tangent line and the point where they cross each other.

ehild
 
  • #3
utkarshakash
Gold Member
855
13
It is not true. z1 and z2 are on the circle which centre is (0,0) z3 can not lie on the circle.

Find the equations of the tangent line and the point where they cross each other.

ehild
I'm not saying that [itex]z_{3}[/itex] will lie on the circle with centre (0,0), Instead the 4 points which I mentioned will lie on some other circle and the centre of that circle will be the mid-point of line joining [itex]z_{3}[/itex] and 0.
 
  • #4
ehild
Homework Helper
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Sorry, I misunderstood you. You are right, z1,z2, (0,0)and z3 are on a circle. Find z3 in terms of z1 and z2. One possible way is to write up the equations of the tangent lines and find the intersection. Do you know the equation of a straight line in term of its normal?


ehild
 
Last edited:
  • #5
utkarshakash
Gold Member
855
13
Sorry, I misunderstood you. You are right, z1,z2, (0,0)and z3 are on a circle. Find z3 in terms of z1 and z2. One possible way is to write up the equations of the tangent lines and find the intersection. Do you know the equation of a straight line in term of its normal?


ehild
I know how to write equation of tangents in terms of x and y but not in the form of z. But I know the equation of a complex line.
 
  • #6
ehild
Homework Helper
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A complex number has x and y components like a vector in a plane. Write up the equations of the tangents in terms of x1,y1 and x2,y2. The coordinates of the point of intersection are x3,y3. You can convert it to the complex number z3=x3+iy3.

ehild
 
  • #7
utkarshakash
Gold Member
855
13
A complex number has x and y components like a vector in a plane. Write up the equations of the tangents in terms of x1,y1 and x2,y2. The coordinates of the point of intersection are x3,y3. You can convert it to the complex number z3=x3+iy3.

ehild
But I have to give the answer in terms of z1 and z2 and converting the answer to the required form will be calculative. Any other methods or should I go with this?
 
  • #8
ehild
Homework Helper
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Do it as you can and we will see.

ehild
 
  • #9
utkarshakash
Gold Member
855
13
Do it as you can and we will see.

ehild
After writing the equation in terms of x and y and then simultaneously solving the two lines I get


[itex] \LARGE z_{3}= \frac{y_{2}-y_{1}}{x_{1}y_{2}-x_{2}y_{1}} + i \frac{x_{1}-x_{2}}{x_{1}y_{2}-x_{2}y_{1}}[/itex]



Now how do I represent this in terms of z1 and z2?
 
  • #10
ehild
Homework Helper
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z=x+iy. x=Re(z) or (z+z*)/2; y=Im(z) or y=(z-z*)/2i. (* means complex conjugate)

ehild
 
  • #11
utkarshakash
Gold Member
855
13
z=x+iy. x=Re(z) or (z+z*)/2; y=Im(z) or y=(z-z*)/2i. (* means complex conjugate)

ehild
After simplification I get

[itex] \LARGE z_{3}= \frac{2(z_{2}-z_{1}- \overline{z_{1}})}{\overline{z_{1}}z_{2}-z_{1} \overline{z_{2}}}[/itex]

But this is not the correct answer.
 
  • #12
ehild
Homework Helper
15,477
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After simplification I get

[itex] \LARGE z_{3}= \frac{2(z_{2}-z_{1}- \overline{z_{1}})}{\overline{z_{1}}z_{2}-z_{1} \overline{z_{2}}}[/itex]

But this is not the correct answer.
Your result is not correct. Check the signs in your derivation.

ehild
 

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