SOLUTION: Consider the parabola y = x^2 a. find the equation of the tangent to the parabola at the point (t, t^2) b. show that the line passing through the focus of the parabola and perp

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Consider the parabola y = x^2 a. find the equation of the tangent to the parabola at the point (t, t^2) b. show that the line passing through the focus of the parabola and perp      Log On


   

Question 1039424: Consider the parabola y = x^2
a. find the equation of the tangent to the parabola at the point (t, t^2)
b. show that the line passing through the focus of the parabola and perpendicular to the tangent in (a) has the equation y = t-2x/4t
c. show that the foot of the perpendicular from the locus to the tangent is the point F(t/2 , 0)
d. find the locus of M, the midpoint of PF
NEED HELP WITH PART C AND D!!!
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


If the foot of the perpendicular to the tangent through the focus is indeed the point , then the zero of the function representing the tangent (i.e. the answer to part a) and the zero of the function representing the perpendicular to the tangent through the focus (i.e. the function given in part b) must both be equal to . I'll let you do the algebra to verify that the zeros are both .

I can't be sure about part d because you don't specify point P. However, I'll go out on a limb and assume it is the focus. In that case, just use the midpoint formulae with the two points and .





So for any given value of , M is the point . Hence, for all real values of the locus of points is:



Assuming, of course, that P represents the focus.

John

My calculator said it, I believe it, that settles it