I am stuck with this question ,:
find the minimum value of where are real numbers with justify your answer
can someone help please?
is the distance squared from the point (-1,0) to (x,y)
So the question is asking the square of the minimum distance from (-1,0) to any point on the line .
i.e. the square of the distance from the point (-1, 0) to the line
you should get the distance to be ,
Actually, your approach is easier.
Now clearly is minimum when , which gives 2.
If you want to know the distance to line approach (don't worry about it, if you haven't learned this):
Rearranging, the line equation is
The slope of the perpendicular line has m = 1
A perpendicular line with m = 1 through point (-1, 0) has the equation
Intersection of the line and the perpendicular line is at (0,1)
So the minimum distance is from (-1,0) to (0,1) which is sqrt(2).
If you are confused on some step, draw a diagram and the graphs.