23. [-/2 Points] DETAILS MY NOTES SCALCET9 3.1.081. For what values of \( a \) and \( b \) is the line \( 4 x+y=b \) tangent to the parabola \( y=a x^{2} \) when \( x=5 \) ? \( a=\square \) \( b= \)

To find the values of \( a \) and \( b \) such that the line \( 4x + y = b \) is tangent to the parabola \( y = ax^2 \) at \( x = 5 \), we start by expressing \( y \) from the line equation: \[ y = b - 4x. \] At the point of tangency, both functions should yield the same \( y \) value when \( x = 5 \): \[ ax^2 = b - 4x. \] Substituting \( x = 5 \): \[ a(5^2) = b - 4(5) \] or \[ 25a = b - 20. \] Now, the derivatives of both expressions must also be equal to ensure tangency. The slope of the line is 4, and the derivative of the parabola is given by: \[ \frac{dy}{dx} = 2ax. \] At \( x = 5 \): \[ 2a(5) = 10a. \] Setting the slopes equal gives: \[ 10a = 4 \] or \[ a = \frac{2}{5}. \] Substituting \( a \) back into the equation \( 25a = b - 20 \): \[ 25\left(\frac{2}{5}\right) = b - 20 \] gives: \[ 10 = b - 20 \] thus \[ b = 30. \] So the values are \( a = \frac{2}{5} \) and \( b = 30 \).