Posted by Anonymous on .
two curves are orthogonal at a point of intersection of their tangents at that point cross at right angles. Show that the curves 2x^2+3y^2=5 and y^2=x^3 are orthogonal at (1,1) and (1,1). Use parametric mode to draw the curves and to show the tangent lines

calculus 
Steve,
By inspection, the curves intersect at (1,1) and (1,1)
ellipse: 4x + 6yy' = 0
y' = 2x/3y
at (1,1) slope = 2/3
at (1,1) slope = 2/3
semicubical parabola: 2yy' = 3x^2
y' = 3x/2y
at (1,1) slope = 3/2
at (1,1) slope = 3/2
The slopes at the intersections are negative reciprocals; hence the curves are orthogonal.