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11 The tangent to the curve y=8x-x^(2) at the point (2,12) cuts the x-axis at the point P. a Find the coordinates of P. b Find the area of the shaded region.

1111 The tangent to the curve y=8x−x2 y=8 x-x^{2} at the point (2,12) (2,12) cuts the x x -axis at the point P P .\newlinea Find the coordinates of P P .\newlineb Find the area of the shaded region.

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Q. 1111 The tangent to the curve y=8x−x2 y=8 x-x^{2} at the point (2,12) (2,12) cuts the x x -axis at the point P P .\newlinea Find the coordinates of P P .\newlineb Find the area of the shaded region.
  1. Calculate Derivative and Slope: Calculate the derivative of y=8x−x2y=8x-x^2 to find the slope of the tangent at the point (2,12)(2,12).dydx=8−2x\frac{dy}{dx} = 8 - 2xAt x=2x=2, dydx=8−2(2)=4\frac{dy}{dx} = 8 - 2(2) = 4.The slope of the tangent line at (2,12)(2,12) is 44.
  2. Write Tangent Line Equation: Write the equation of the tangent line using point-slope form.\newliney−y1=m(x−x1)y - y_1 = m(x - x_1)\newliney−12=4(x−2)y - 12 = 4(x - 2)
  3. Find X-Intercept: Find where this tangent line intersects the x-axis by setting y=0y=0.\newline0−12=4(x−2)0 - 12 = 4(x - 2)\newline−12=4x−8-12 = 4x - 8\newline4x=44x = 4\newlinex=1x = 1

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