Extension 2 Patel Exercise 4O Question 6,19 (1 Viewer)

[juampabonilla]

Question 6: P is a variable on the line x=4 and OPQR is a rectangle in which the length of OP is twice the length of OR. Find the locus of S in Cartesian form, where S is the point of intersection of the diagonals.

Question 19: Let z=x+iy and w= u+iv =(z-1)^2 +2 be complex numbers in an Argand Diagram. Show that as z moves along the y-axis from O(0,0) to A(0,2) the point w moves along an arc of a certain parabola. Find the corresponding points on this parabola and the cartesian equation of this parabola.

I would appreciate any help.

 

[Axio]

For 19 (I think this is right): sub z=x+iy into w=(z-1)^2 +2 and expand. The sub in x=0 into that to obtain a parabola.

 

[WhoStanLeee]

Question 6: P is a variable on the line x=4 and OPQR is a rectangle in which the length of OP is twice the length of OR. Find the locus of S in Cartesian form, where S is the point of intersection of the diagonals.

Question 19: Let z=x+iy and w= u+iv =(z-1)^2 +2 be complex numbers in an Argand Diagram. Show that as z moves along the y-axis from O(0,0) to A(0,2) the point w moves along an arc of a certain parabola. Find the corresponding points on this parabola and the cartesian equation of this parabola.

I would appreciate any help.

Corresponding points:

Cartesian Equation:

 

[braintic]

Question 6:
View attachment Patel Ex 4O Q6.pdf

 

[braintic]

Corresponding points:

Cartesian Equation:

I'm afraid you haven't SHOWN it is a parabola, you've ASSUMED it is a parabola.

 

[WhoStanLeee]

The question states that w moves along a certain parabola. As the question asked for the cartesian equation of the parabola, is y^2 = -4(x-3) not sufficient? My assumption is valid because I am told it is a parabola?

 

[InteGrand]

Pretty sure we can assume an equation of the form describes a parabola...

Just like how, for example, we are allowed to assume that represents a circle.

 

[braintic]

Pretty sure we can assume an equation of the form describes a parabola...

Just like how, for example, we are allowed to assume that represents a circle.

I don't think you understood the point of my comment.

That solution merely fitted a parabola to a few points. That doesn't necessarily mean that the locus is a parabola. It just means that the parabola he has found happens to go through some points that lie on the locus.

Of course his answer is the locus, but it is not proven.