erisedk said:Homework Statement
Find the minimum distance between the curves y^2 - xy - 2x^2 = 0 and y^2 = x - 2.
Ans: 7/(4*root2)
Homework Equations
The Attempt at a Solution
It's supposed to be done using concepts from parabolas. I tried using calc but it got convoluted.
A parabola is a type of curve that is formed by the graph of a quadratic equation, y = ax^2 + bx + c, where a, b, and c are constants.
To find the minimum distance between two parabolas, we can use the distance formula. We first set the two parabolas equal to each other to find the points of intersection. Then, we can use the distance formula to find the distance between these points, which will give us the minimum distance between the two parabolas.
The number 7 indicates the y-coordinate of the vertex of the parabola. In the equation y = ax^2 + bx + c, the vertex has coordinates (-b/2a, c - b^2/4a). Since there is no x-term in the given equation, the x-coordinate of the vertex is 0, and the y-coordinate is 7.
If the two parabolas have different coefficients for their x^2 terms, then they will intersect at two points. If the coefficients are the same, then the two parabolas are either identical or parallel and will not intersect. This can be determined by setting the two equations equal to each other and solving for x.
Yes, the minimum distance between two parabolas can be 0 if the two parabolas are identical. In this case, they would intersect at every point and have a minimum distance of 0.