If the tangent to the curve y = x ^ 3 - x ^ 2 + x at the point (a,...

Given,

Curve $y = 5{x}^{2} + 2x - 25$

Now differentiating to find slope at point $P\left(2, - 1\right) we get,</p><p>${y}^{'} = 10x + 2 ⇒ {y}_{P}^{'} = 22$</p><p>Now tangent to curve at$P$will be,</p><p>$y + 1 = 22\left(x - 2\right)$</p><p>$⇒ y = 22x - 45$</p><p>Now the tangent is also tangent to the curve$y = {x}^{3} - {x}^{2} + x$and passing through \left(a, b\right)$ so plottnig the diagram we get,

Now differentiating the curve $y = {x}^{3} - {x}^{2} + x$ we get,

$\dfrac{dy}{dx} = 3{x}^{2} - 2x + 1$

${\dfrac{dy}{dx}}_{Q = \left(a, b\right)} = 3{a}^{2} - 2a + 1$

So, equating with tangent slope we get,

$3{a}^{2} - 2a + 1 = 22$

$⇒ 3{a}^{2} - 2a - 21 = 0$

$⇒ 3{a}^{2} - 9a + 7a - 21 = 0$

$⇒ \left(3a + 7\right)\left(a - 3\right) = 0$

Now from curve $b = {a}^{3} - {a}^{2} + a$

So, at $a = 3 ⇒ b = 21$

Also $a = 3 \text{\text{ and } }$ b = 21 are satisfying the line $y = 22x - 45$

So, \left|2a + 9b\right| = \left|2 \times 3 + 9 \times 21\right| = 195$</p><p>Note at$a = - 7 / 3 tangent will be parallel so it is rejected