Geometry

The equation of the curve is . To find the condition when the line is normal to the curve, we need to find the slope of the tangent line at a point on the curve and the slope of the normal line.
Step 1: Find the slope of the curve.
Implicitly differentiate the equation of the curve: Thus, the slope of the tangent at any point on the curve is .
Step 2: Find the slope of the normal.
The slope of the normal line is the negative reciprocal of the slope of the tangent. Therefore, the slope of the normal line is: Step 3: Find the slope of the given line.
The equation of the given line is , which can be rewritten as: Step 4: Condition for the line to be normal to the curve.
For the line to be normal to the curve, the product of the slopes of the tangent and the normal must be . Thus, we set: Since , we substitute into this equation: Thus, the correct condition is . Hence, the correct answer is:

Step 1: Check the equation of the line. The equation of the line is . We need to check which side of this line the points and lie on.
Step 2: Calculate the value of for each point. For the point : For the point : Both values of are positive, meaning both points lie on the same side of the line. Thus, the correct answer is:

The first equation represents a hyperbola, and the second equation represents an ellipse. To find the number of intersection points, we need to solve the system of equations by substituting one into the other and analyzing the resulting equation.
After solving the system, we find that there are 2 points of intersection.
Thus, the correct answer is .

Step 1: Examine the first part of the function. For , the function is: Taking the derivative: Setting to find critical points: Thus, the critical points are and . For , , indicating a local minimum.
For , , indicating a local maximum.
Step 2: Examine the second part of the function. For , the function is: The derivative is: Setting gives , which occurs at . This is a local maximum since has a maximum at .
Step 3: Examine the third part of the function. For , the function is: This function is always decreasing, so no maximum exists here.
Step 4: Conclusion. The function has a local maximum at for the first piece, but it is not the global maximum because the global maximum occurs at for the second piece of the function, where attains its maximum value. Thus, the correct answer is: