Pranav-Arora said:Homework Statement
Prove that the locus of middle points of all chords of the parabola ##y^2=4ax## which are drawn through the vertex is the parabola ##y^2=2ax##.
Homework Equations
The Attempt at a Solution
The mid-point of the line joining the vertex and a point ##(h,2\sqrt{ah})## on parabola is ##(h/2,\sqrt{ah})## but what am I supposed to do with this?
I like Serena said:
A locus on a parabola refers to the set of all points on the parabola that satisfy a specific condition or property. In this case, the locus is the set of middle points of chords on the parabola.
The locus of middle points of chords on a parabola can be proven using the distance formula. This involves finding the distance between the midpoint of a chord and the focus of the parabola. If the distance is equal to the distance between the midpoint and the directrix of the parabola, then the point lies on the locus.
Proving the locus of middle points of chords on a parabola allows us to better understand the properties of the parabola and its relationship to its focus and directrix. It also has practical applications in fields such as engineering and physics.
Yes, the locus of middle points of chords on a parabola can be used to find the equation of the parabola. This is because the locus is a set of points that satisfy a specific condition, which can be translated into an algebraic equation.
Yes, besides using the distance formula, there are other methods to prove the locus of middle points of chords on a parabola. These include using the properties of the focus and directrix, as well as using the definition of a parabola as the locus of points that are equidistant from a fixed point and a fixed line.