Hello, Flay!

Parametric equations: ., a variable chord of the parabola , subtends a right angle at the vertex

a) If are parameters corresponding to and , show that:

The points are: .

The slope of is: .

The slope of is: .

Since , we have: .

Note: .Since , point becomes: .

The slope of the tangent is: .b) If the tangents at intersect at , find the locus of

At , we have the point and slope

The equation of the tangent at is:

. .

At , we have the point and slope

The equation of the tangent at Q is:

. .

The two tangents intersect: .

. . Factor: .

Substitute into [1]: .

The locus of is a horizontal line!

We have: .c) If is the midpoint of chord , determine the locus of

The midpoint has coordinates:

. .

Square [3]: .

Substitute [4]: .

The locus of is another parabola,

. . somewhat "narrrower" with vertex (0,4).