How to find the equations?

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1 Answer
Feb 25, 2018

Let the line L passing through the point (1, 7) touches the parabola
#y=-3x^2+5x+2# at the point(a, b), line L is tangent to the parabola at (a, b), so we have:
#-3a^2+5a+2=b# => eq-1

#slope=m=(b-7)/(a-1)#
the parabola has the same slope at the point(a, b):
#y'=-6x+5#
#y'(a)=-6a+5#
#(b-7)/(a-1)=-6a+5# => eq-2

solving the eq-1 and eq-2 simultaneously will give us:
#a=0,b=2 or a=2, b=0#
thus the equation of the line is:
#y-7=(2-7)/(0-1)(x - 1)#
#y-7=5(x-1)#
#y=5x+2#

and for the 2nd point:
#y-7=(0-7)/(2-1)(x - 1)#
#y-7=-7(x-1)#
#y=-7x+14#

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