Question

Find the equations of the tangent and normal to each of the following conics, the lengths of the subtangent and subnormal, then trace the curve showing these lines.

y = x2 – 6x + 4 at (4, -4)

Answer 1

tangent line:

$y-y_0=f'(x_0)(x-x_0)$

$f'(x)=2x-6$

$f'(4)=2$

$y+4=2(x-4)$

$y=2x-12$

normal line:

$y-y_0=-(x-x_0)/f'(x_0)$

$y+4=-(x-4)/2$

$2y+x=-4$

subtangent is the projection of the segment of the tangent onto the x-axis

subnormal is the projection of the segment of the normal onto the x-axis

A segment of a tangent to a curve lying between the tangency point (the point at which a tangent is drawn to a curve) and the intercept of the tangent with the x-axis is called the length of the tangent.

A segment of a line normal to a tangent lying between the tangency point and the intercept of the normal with the x-axis is called the length of the normal.

length of the subtangent: $6-4=2$

length of the subnormal: $4+4=8$