Hello - there are a few ways - from your description (separate curve), probably the simplest is to make a line, then Match the line back to the curve for tangency, with the other end held at Position.

For a single curve use Extend > ToPoint with Type= Arc or Smooth.

The key that I might have missed to explicitly state is that I don’t want to modify either of the existing curves or lines. I want to add a new line and find new points along the curve.

There is a major limitation on where the end point of the line can be due to geometry. The end point of the line must be on the surface created by sweeping a line tangent to the curve along the curve. If the curve is planar then the surface will be a plane. If the curve is not planar the surface will be curved.

Two methods to create a line from the end point to the curve assuming the end point satisfies the condition above…

Method A
Line command and select Tangent option.
Select the end point (not the point on the curve)
Move the cursor to the curve. A line should appear between the end point and tangent to the curve. Left click to create the line.

Method B
Make sure the Tan OSnap is on.
Line command
Select the end point (not on the curve)
Move the cursor near the curve and move along the curve until the Tan OSnap appears.
Left click to create the line.