At what points does the curve r(t) = ti + (5t − t2)k intersect the paraboloid z = x2 + y2? (if an answer does not exist, enter dne.)

Answer : curve r(t) = t i + (2t - t 2)k intersect the paraboloid z = x 2 + y 2

By the equation of r(t), x = t, y = 0, and z = .  

Thus, z =  

on solving we get,

=  

Now, we can solve for t  

= 0

So, = 0

2t ( t - 5/2) = 0  

t = 0 , t = 5  

plugging these values in t = 0 into r(t)

r(0) = <0 , 0 , 0>

r(5) = < 5, 0 , 25 >

The curve intersect the paraboloid at and

Further explanation:

Given:

The equation of the curve

The equation of the paraboloid

Explanation:

From theequation of the curve

The value of is and the value of is and the equation of is

Substitute for , for and in equation of the paraboloid

The value of can be obtained as follows,

The value of is and the value of z can be obtained as follows,

The curve intersect the paraboloid at and

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Intersecting curves

Keywords: Points, curve, intersects, paraboloid, not, exist, , at what.

its a straight line parallel to the time axis

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