PQ is a double ordinate of a parabola. Find the locus of its point of trisection.

To find the locus of the point of trisection of the double ordinate PQ of a parabola, let's start with the basic properties of a parabola. Consider the standard form of a parabola given by the equation:

Parabola Basics

The equation of a parabola that opens upwards can be expressed as:

y² = 4ax

Here, 'a' is the distance from the vertex to the focus.

Understanding Double Ordinates

A double ordinate is a line segment perpendicular to the axis of symmetry of the parabola, intersecting it at two points. If we take a double ordinate at a distance 'h' from the vertex, the coordinates of points P and Q can be represented as:

• P(h, 2√ah)
• Q(h, -2√ah)

Finding the Trisection Point

The point of trisection divides the segment PQ into three equal parts. The coordinates of the trisection point T can be calculated as:

T = (h, (2√ah)/3)

Determining the Locus

To find the locus of point T as 'h' varies, we need to express 'h' in terms of 'y'. From the equation of the parabola:

y² = 4ax

We can solve for 'h' in terms of 'y':

h = (y²)/(4a)

Substituting to Find the Locus

Now, substituting 'h' back into the coordinates of T gives:

T = ((y²)/(4a), (2√a(y²)/(4a))/3)

After simplifying, we find:

T = ((y²)/(4a), (y²)/(6√a))

Final Equation of the Locus

To eliminate 'h' and find the relationship between x and y, we can express 'x' in terms of 'y':

x = (y²)/(4a}

Thus, the locus of the point of trisection T can be expressed as:

y² = 24ax

This equation represents another parabola, indicating that the locus of the point of trisection of the double ordinate of a parabola is itself a parabola.