Question

If the tangent to the curve y = x3 - x2 + x at the point (a, b) is also tangent to the curve y = 5x2 + 2x - 25 at the point (2, -1), then |2a + 9b| is equal to _____ .

Answer 1

Slope of tangent to curve y = 5x2 + 2x – 25,
$m=\left(\frac{dy}{dx}\right)_{at(2,−1)}=22$
Equation of tangent: y + 1 = 22(x – 2)
y = 22x – 45
Slope of tangent to y = x3 – x2 + x at point (a, b) = 3a2 – 2a + 1
3a2 – 2a + 1 = 22
3a2 – 2a – 21 = 0
$∴a=3$ or $a=−\frac{7}{3}$
Also b = a3 – a2 + a
Then $(a,b)=(3,21)$ or, $(−\frac{7}{3},–\frac{151}{9})$
$(−\frac{7}{3},–\frac{151}{9})$ does not satisfy the equation of tangent
$∴a=3,b=21$
the, $|2a+9b|=|2\times3+9\times21| =|195| = 195$
So, the answer is 195.