Question

How to find the equations?

Answer 1

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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