Question

(1 point) find a vector function that represents the curve of intersection of the paraboloid z=8x2+3y2z=8x2+3y2 and the cylinder y=5x2y=5x2. use the variable t for the parameter.

Answer 1

The vector function that represents the curve of intersection of the given paraboloid and cylinder is,

`\vec{r}=<t,5t^2,8t^2+75t^4>`

Solution-

Here we have to calculate the vector function represents the curve of intersection of the paraboloid '' z = 8x²+3y² '' and the cylinder " y=5x² "

All are in rectangular form and we have to convert them into parametric form.

`Let \ x =t,`

putting it in the equation of the cylinder equation we can get the value of y, so

`y = 5x^2 = 5t^2`

Now, we have the values of x and y in parametric form. Putting the values of x and y in the paraboloid equation, we can get the value of z, so

`z = 8x^2+3y^2= 8t^2+3(5t^2)^2 = 8t^2+75t^4`

∴ Vector function that represents the curve of intersection is,

`\vec{r}=<t,5t^2,8t^2+75t^4>`