If the tangent to the curve y=x^3-x^2+x at the point (a, b) is also tangent to the curve y = 5x^2 + 2x - 25 at the point (2, -1), then |2a + 9b| is equal to __________.

Slope of tangent to curve y=5 x^2+2 x-25 \ =m=((d y / d x))_ at(2,-1)=22 \ Therefore Equation of tangent: y+1=22(x-2) \ Therefore y=22 x-45 \ \ Slope of tangent to y=x^3-x^2+x at point (a, b) \ =3 a^2-2 a+1 \ \ 3 a^2-2 a+1=22 \ 3 a^2-2 a-21=0 \ Therefore a=3 or -(7 / 3) \ \ Also b=a^3-a^2+a \ Then (a, b)=(3,21) or (-(7 / 3),-(151 / 9)) \ \ (-(7 / 3),-(151 / 9)) does not satisfy the equation of tangent \ Therefore a=3, b=21 \ Therefore |2 a+9 b|=195

If the tangent to the curve yx3x2x at the point a b is ... | null (Mathematics)

\begin{aligned} & \text{Slope of tangent to curve } y=5 x^{2}+2 x-25 \\ & =m=\left(\frac{d y}{d x}\right)_{\mathrm{at}(2,-1)}=22 \\ & \therefore \quad \text{Equation of tangent}: y+1=22(x-2) \\ & \therefore \quad y=22 x-45 \\ \\ & \text{Slope of tangent to } y=x^{3}-x^{2}+x \text{ at point } (a, b) \\ & =3 a^{2}-2 a+1 \\ \\ & 3 a^{2}-2 a+1=22 \\ & 3 a^{2}-2 a-21=0 \\ & \therefore \quad a=3 \text{ or } -\frac{7}{3} \\ \\ & \text{Also } b=a^{3}-a^{2}+a \\ & \text{Then } (a, b)=(3,21) \text{ or } \left(-\frac{7}{3},-\frac{151}{9}\right) \\ \\ & \left(-\frac{7}{3},-\frac{151}{9}\right) \text{ does not satisfy} \\& \text{the equation of tangent} \\ & \therefore \quad a=3, b=21 \\ & \therefore \quad |2 a+9 b|=195 \end{aligned}

Explanation

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