Answer-
The vector function that represents the curve of intersection of the given paraboloid and cylinder is,
[tex]\vec{r}=<t,5t^2,8t^2+75t^4>[/tex]
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.
[tex]Let \ x =t,[/tex]
putting it in the equation of the cylinder equation we can get the value of y, so
[tex]y = 5x^2 = 5t^2[/tex]
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
[tex]z = 8x^2+3y^2= 8t^2+3(5t^2)^2 = 8t^2+75t^4[/tex]
∴ Vector function that represents the curve of intersection is,
[tex]\vec{r}=<t,5t^2,8t^2+75t^4>[/tex]
