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# The line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P. The x-coordinate of P is –2

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The line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P. The x-coordinate of P is –2. Determine the value of a + b without using calculus.

May 29, 2023

#1
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Here's how you can solve this problem without using calculus:

Since the line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P, the line and the parabola share a common point. This means that the line's equation can be written in the form of a parabola's equation.

The equation of a parabola in vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

The line y = –11x – 7 can be written in the form of a parabola's equation by completing the square. Completing the square gives us y = –11(x + 1)^2 + 2.

The vertex of the parabola y = –11(x + 1)^2 + 2 is (-1, 2).

Since the line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P, and the x-coordinate of P is –2, we know that the point P is (-2, 2).

Since the point P is (-2, 2), we know that a + b = 2.

Therefore, the value of a + b is 2.

May 29, 2023
#2
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I think this is wrong because point P is (-2, 15), not (-2, 2). Thank you for your answer though.

somebodyIUsedToKnow  May 29, 2023