SOLUTION: Find the cooordinates of the two points on the curve y=4-x^2 whose tangents pass through the point (-1,7)

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Question 978015: Find the cooordinates of the two points on the curve y=4-x^2 whose tangents pass through the point (-1,7)
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
The line minus the parabola should have ONE solution. Each of these lines must have slope -2x, using derivative for the given parabola equation.

y-7=%28-2x%29%28x%2B1%29, equation for the tangent lines in point-slope form.
y=-2x%5E2-2x%2B7-----Again this is a TANGENT LINE, not a parabola. We do not have yet any value for the slope.
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This line must intersect the parabola in ONLY ONE POINT. Their difference must be zero for only one value of x.

-2x%5E2-2x%2B7-%284-x%5E2%29=0, difference between line and parabola.
-2x%5E2-2x%2B7-4%2Bx%5E2=0
-x%5E2-2x%2B3=0
x%5E2%2B2x-3=0
%28x-1%29%28x%2B3%29=0
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One value where tangent on the parabola is x=1; and the other value where tangent on parabola is x=-3.

FIND CORRESPONDING y VALUES ON PARABOLA
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y=4-%281%29%5E2
y=4-1
y=3
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y=4-%28-3%29%5E2
y=4-9
y=-5
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The points where line is tangent to parabola and tangent includes (-1,7) are (1,3) and (-3,-5).

You can make a sketch, graph on your own to be more certain.