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

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Solution

let $C_{1} : y = x^{3} - x^{2} + x$ & $C_{2} : y = 5 x^{2} + 2 x - 25$

$\frac{d y_{1}}{d x} = 3 a^{2} - 2 a + 1$ at $(a, b)$

$\frac{d y_{2}}{d x} = 10 x + 2$ = 22 at $(2, - 1)$

$∴ 3 a^{2} - 2 a - 21 = 0, a = 3$ , $a = - 7 / 3$
$a = - 7 / 3$ (rejected)

for $a = 3, b = 21$

$∴ | 2 a + 9 b | = 195$

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a) F i n d t h e p o i n t s o n t h e c u r v e y = x^{3} - 10 x + 8 a t w h i c h t h e \tan g e n t i s p a r a l l e l t o t h e l i n e y = 2 x + 1 b) I s t h e g i v e n l i n e y = 2 x + 1 \tan g e n t t o t h e c u r v e ? W h y ?

y = b e^{- x / a}

a

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