There exists a constant c so that among all chords of the parabola passing through
is a fixed constant. Find the constant
To get a handle on the constant we can look at some particular cases.
Suppose we let approach a vertical line. Then approaches 0, and approaches so approaches Hence,
Now, suppose we take and Then
Hence, so and Therefore, (Note that this makes the focus of the parabola.)
For a full solution, let's check that this value works. Let be the equation of line Setting we get
or Let and be the roots of this equation. By Vieta's formulas, and
Also, and are and in some order, so
We have that and
Hence,
To get a handle on the constant we can look at some particular cases.
Suppose we let approach a vertical line. Then approaches 0, and approaches so approaches Hence,
Now, suppose we take and Then
Hence, so and Therefore, (Note that this makes the focus of the parabola.)
For a full solution, let's check that this value works. Let be the equation of line Setting we get
or Let and be the roots of this equation. By Vieta's formulas, and
Also, and are and in some order, so
We have that and
Hence,