Question

Determine whether the line x - 2y + 3 = 0 is tangent to the parabola y^2 = 16x.

Answer 1

To determine whether the line x - 2y + 3 = 0 is tangent to the parabola y^2 = 16x, we need to check if the line intersects the parabola at a single point.

Step-by-step explanation:

To determine whether the line x - 2y + 3 = 0 is tangent to the parabola y^2 = 16x, we need to check if the line intersects the parabola at a single point.

We can do this by substituting the equation of the line into the equation of the parabola:

• Substitute x = 2y - 3 into y^2 = 16x
• Simplify the equation
• If the equation simplifies to y^2 = 16x, then the line is tangent to the parabola. Otherwise, the line intersects the parabola at two points and is not tangent.

Let's go through the steps:

1. Substitute x = 2y - 3 into y^2 = 16x: y^2 = 16(2y - 3)
2. Simplify the equation: y^2 = 32y - 48
3. This quadratic equation can be written in the form y^2 - 32y + 48 = 0.
4. Since this equation is not y^2 = 16x, the line x - 2y + 3 = 0 is not tangent to the parabola y^2 = 16x.