Find a vector function that represents the curve of intersection of the paraboloid z=4x2+3y2
and the cylinder y=5x2
. Use the variable t for the parameter.
r(t)=
To find the curve of intersection between the paraboloid and the cylinder, we need to find the values of x, y, and z that satisfy both equations.
First, let's substitute y=5x^2 into the equation of the paraboloid:
z = 4x^2 + 3(5x^2)
z = 4x^2 + 15x^2
z = 19x^2
Now, let's express x and z in terms of the parameter t:
x = t
z = 19t^2
To find y, substitute x=t into the equation of the cylinder:
y = 5(t^2)
So, the vector function that represents the curve of intersection is:
r(t) = <t, 5t^2, 19t^2>
To find a vector function that represents the curve of intersection between the paraboloid and the cylinder, we need to find the values of x, y, and z as functions of t.
First, we can rewrite the equation of the cylinder as:
y = 5x^2
Substituting this expression for y into the equation of the paraboloid, we get:
z = 4x^2 + 3(5x^2)
z = 4x^2 + 15x^2
z = 19x^2
Now we have x and z expressed in terms of each other. To find y, we can substitute the expression for x from the equation of the cylinder into the equation for y:
y = 5( x^2 )
Now we have expressions for x, y, and z in terms of the parameter t:
x = t
y = 5t^2
z = 19t^2
Therefore, the vector function representing the curve of intersection is given by:
r(t) = <t, 5t^2, 19t^2>