Answer the questions given below, which concern the geometric ideas of tangent and normal for a curve represented by an equation in x and y.
a. (i) Define the tangent line to a plane curve at a point on the curve.
b. (ii) For a curve expressed by an equation F(x,y)=0, describe in general terms how the slope of the tangent at a point (x0,y0) is obtained.
c. (iii) Give a reason why, provided the tangent at (x0,y0) is neither vertical nor horizontal, the slope of the normal at that point is the negative reciprocal of the slope of the tangent.
d. (iv) State the condition on dxdy at a point that makes the tangent to the curve perpendicular to the x–axis at that point.