Applications Of Integrals

To find the point on the parabola that is closest to the line , we need to use the concept of the distance between a point and a line in the coordinate plane.

1. Identify the general point on the parabola. The parabola is given by the equation . Therefore, any point on the parabola can be represented as .
2. Find the distance from point to the line . The formula for the distance from a point to a line of the form is:

1. For the line , rearrange it into standard form . Here, , , and .
2. Substitute the point into the distance formula:

1. To minimize the distance, minimize the numerator . This is equivalent to minimizing the expression inside the absolute value, .
2. Set . To find the critical points, compute the derivative and set it to zero:

1. Set :

1. Substitute back into the parabola's equation to find the -coordinate of the point:

1. Thus, the point is the point on the parabola closest to the line.
2. Verify by considering the options given: Among the options , , , and , only is found to be the point closest to the line based on the calculations.

Therefore, the correct answer is (2, 8).

Given the quadratic curve , we know it passes through the point (1, 2) and has a tangent line at the origin with the equation . We need to determine the values of , , and .

1. Since the curve passes through the point (1, 2), we substitute these values into the equation of the curve: (Equation 1)
2. The tangent at the origin is , which implies that the derivative of the curve at the origin should equal 1 (the slope of the line ). Differentiating the curve equation

1. , we get: Evaluate this at : (Equation 2)
2. Substitute the value of from Equation 2 back into Equation 1: (Equation 3)
3. Now, evaluate the options using Equations 2 and 3 to find which set of values satisfies both equations:
   • Option A:
     Substituting into the equations: Equation 2 is not satisfied since . Hence, Option A is incorrect.
   • Option B:
     Substituting into the equations: Equation 2 is not satisfied since . Hence, Option B is incorrect.
   • Option C:
     Substituting into the equations: Equation 1: ✅
     Equation 2: ✅
     Equation 3: ✅
     This option satisfies all conditions.
   • Option D:
     Substituting into the equations: Equation 3 is not satisfied since . Hence, Option D is incorrect.

Thus, the correct values of , , and that satisfy all the conditions are given in Option C: .