If the slope of parabola 𝑦=𝑎𝑥2+𝑏𝑥+𝑐y=ax 2 +bx+c, where 𝑎,𝑏,𝑐∈a,b,c∈𝑅R \{0}{0} at points (3,2)(3,2) and (2,3)(2,3) are 3737 and 2020 respectively, then find the value of 𝑎a.

Question

If the slope of parabola 𝑦=𝑎𝑥2+𝑏𝑥+𝑐y=ax 2 +bx+c, where 𝑎,𝑏,𝑐∈a,b,c∈𝑅R {0}{0} at points (3,2)(3,2) and (2,3)(2,3) are 3737 and 2020 respectively, then find the value of 𝑎a.

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Solution

The slope of a parabola y = ax^2 + bx + c at any point (x, y) is given by the derivative of the function, which is 2ax + b.

Given that the slopes at points (3,2) and (2,3) are 37 and 20 respectively, we can set up the following equations:

2a3 + b = 37 2a2 + b = 20

This is a system of linear equations. We can solve it by subtracting the second equation from the first to eliminate b:

2a3 - 2a2 = 37 - 20 2a = 17

Then, divide both sides by 2 to solve for a:

a = 17/2 = 8.5

So, the value of a is 8.5.

This problem has been solved

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