Find the point P in the first quadrant that lies on the hyperbola 7y^2-3x2=6 and is closest to the point A(90,0). If we write the point as P(a,b), then a =? b=?

Question

need help please

Answer 1

Let (x,y) be a point on the hyperbola.  Use the distance formula to find the distance between (x,y) and (90,0).

 

This will give sqrt((90-x)^2+y^2).  Note that 7y^2-3x^2=6 since (x,y) lies on the hyperbola.  Solve for y^2 to get y^2=6/7+(3/7)x^2. 

 

Plug this in for y^2 in the distance formula: d=sqrt((90-x)^2+6/7+(3/7)x^2).  Minimize this function by first finding the critical points.  Then do the First Derivative test to find the local min.