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

Correct Answer: 195

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.