Let c1 be the intersection curve of the paraboloid z x2 y2...

Let C1 be the intersection curve of the paraboloid z = x^2 + y^2 and the plane z = 2x. Let C2 be the circle parameterized by r2 (t) = cos(t)i + sin(t)j + k, t ∈ [0, 2π]. a) Give a parameterization of the curve C1. b) Find all the points of intersection of curves C1 and C2. c) Show that the point (1, -1, 2) is located on the curve C1, then give a parameterization of the line tangent to C1 at this point.

Transcript

00:01 We have given in this question c1 be the intersection curve on a parabola z is equal to we have given x square plus y square and on the plane z is equal to we have given is equal to two x so we can say if z is equal to x square plus y square and z is equal to two x so here we can say the x square plus y square is equal to two x if x square plus y square is equal to two x then put the value of x minus one is equal to quote t and here put the value of y is equal to sine t sign t so we can say 2 is equal to 2x is equal to 2 multiply by the value of x from here 1 plus code t that means it becomes 1 plus cos t this is 1 plus cost this is also 1 plus cost t and this is sine t so we can say the permittization of this becomes r2 t is equal to t is equal to t is even to cost t plus 1 cost t plus 1 i scale plus sine t sign t sign t j scale plus 2 multiply by that mean is to value of 2 it becomes 1 plus cost t 1 plus cot t cost t this is cost t also and the here multiply we can say k space and here we have given the value of 1 less than and equal 0 and the value of 1 less than and equal 2 pi the value of 1 less than and equal 2 pi and greater than and equal 0 this is answer we have say the permittization by this so this is our answer and we are going to question number 2nd and question number 2nd we have to find find the points of intersection curve c and c2 so c1 and c2 we have to find the curve so intersection of c1 intersection of c1 is equal to x minus one whole of square plus y square is equal to one and and x square plus y square is equal to one so we can say here it becomes x minus one of square plus y square x minus one whole of square plus y square is equal to also x square plus y square so we can cancel out x square y square from here x becomes x minus one becomes is equal to zero and if x minus one equal to zero so we can say it becomes two x is equal to one two x is equal to one if two x is equal to one by two this is the first point and secondly we know that x square plus y square is equal to 1 that mean y square…

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