Answer to Question #270946 in Analytic Geometry for Aliyah

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: $4+4=8$