If 2x + y + k = 0 is a normal to the parabola y2 = –16x, then find the value of k.

To find the value of k, we need to determine the equation of the tangent line to the parabola y^2 = -16x at a given point.

Step 1: Find the derivative of the parabola equation. Differentiating y^2 = -16x with respect to x, we get: 2y(dy/dx) = -16

Step 2: Simplify the derivative equation. Dividing both sides by 2y, we have: dy/dx = -8/y

Step 3: Determine the slope of the tangent line. The slope of the tangent line is given by the derivative evaluated at the point of tangency. In this case, the normal line 2x + y + k = 0 is perpendicular to the tangent line. Therefore, the slope of the tangent line is the negative reciprocal of the slope of the normal line, which is 2.

Step 4: Set up the equation for the slope of the tangent line. Using the slope-intercept form of a line, y = mx + c, where m is the slope and c is the y-intercept, we have: dy/dx = -8/y -8/y = 2

Step 5: Solve for y. Cross-multiplying, we get: -8 = 2y y = -4

Step 6: Substitute the value of y into the equation of the parabola to find the corresponding x-coordinate. Using the equation y^2 = -16x, we have: (-4)^2 = -16x 16 = -16x x = -1

Step 7: Determine the equation of the tangent line. Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we have: y - (-4) = 2(x - (-1)) y + 4 = 2(x + 1) y + 4 = 2x + 2 y = 2x - 2

Step 8: Find the value of k. Since the normal line 2x + y + k = 0 is perpendicular to the tangent line, the slopes of the two lines multiply to give -1. Therefore, we have: 2 * 2 = -1 4 = -1 This is not possible, so there is no value of k that satisfies the given conditions.