Question

Find the shortest distance from the point (0,6) to the parabola x² - 4y = 0. (Give an exact answer. Use symbolic notation and fractions where needed.) shortest distance: ________

Answer 1

The shortest distance from the point (0,6) to the parabola x² - 4y = 0 is 24√17 / 17.

Step-by-step explanation:

To find the shortest distance from the point (0,6) to the parabola x² - 4y = 0, we can use the formula for the distance between a point and a curve. The formula is given by d = |ax + by + c| / √(a² + b²).

For the parabola x² - 4y = 0, we have a = 1, b = -4, and c = 0. Plugging these values into the formula, we get d = |x - 4y + 0| / √(1² + (-4)²) = |x - 4y| / √17.

Substituting the coordinates of the point (0,6), we have d = |0 - 4(6)| / √17 = 24 / √17 = 24√17 / 17.

Answer 2

To find the shortest distance from the point (0,6) to the parabola x² - 4y = 0, we minimize the distance function by substituting the y from the parabola into it, differentiating, solving for x, finding y, and then calculating the actual distance using these coordinates.

Step-by-step explanation:

The question asks to find the shortest distance from the point (0,6) to the parabola x² - 4y = 0. The equation of the parabola can be rewritten as y = (1/4)x² to make it easier to work with. To find the shortest distance, we need to minimize the distance function, which in this case is the square root of ((x-0)² + (y-6)²). However, for minimization, we can instead minimize the square of the distance to avoid the square root and then find the minimum point using calculus.

The minimization process involves substituting the y from the parabola's equation into the distance formula, getting a function in terms of x, and then finding its derivative. After setting the derivative equal to zero, we can solve for the x-coordinate of the closest point on the parabola to (0,6). Substituting this x back into the parabola's equation yields the y-coordinate. Finally, we calculate the actual distance using these coordinates.