Please help me with this calc problem!

Question

 Chord AB joins A(-4,0) and B(0,2) on the graph of y^2=4+x

 

1.At what point(s) is the tangent to the curve parallel to AB?

2.At what point(s) does the perpendicular bisector of AB intersect the curve?

 

 

I got (0,-2) and (0,2) for question 1 but I'm not sure if that's right, and I am totally clueless for question 2. Please help!!

Answer 1

Hi Linda,

 

This question is really best answered with a video so that you can see what is happening. If I get a chance I will update with a video link later.

 

For now for part 1:

You are looking for the point on the opening to the right parabola where there is a tangent line that is parallel to the segment joining (-4,0) and (0,2). This is actually an application of the Intermediate Value Theorem. What is the slope of the segment? At what point in the interval from -4 to 0 is the first derivative of the "function" equal to the slope of the segment? I will tell you that the answer is (-3,1). 

 

For part 2:

Find the equation of the perpendicular bisector of the segment. That line will have a slope that is the negative reciprocal of the slope of the segment and it will pass through its midpoint. Consider using the point-slope formula for the equation of the line after you have determined those values. Once you have the equation of that line, you need to determine the points of intersection of that line with the original parabola; solve a system of two equations in two unknowns. That will involve solving a quadratic equation using the quadratic formula.

 

If in fact I see a response from you that you would really like to see a video solution I will take the time to do so.