Question

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

Answer 1

$ y=5 x^2+2 x-25 P(2,-1) $
$y^{\prime}=10 x+2 $
$ y_P^{\prime}=22 $
$ \therefore \text { tangent to curve at } P $
$ y+1=22(x-2) $
$ y=22 x-45$

$ \left.\frac{ dy }{ dx }\right|_{ C _2}=3 x ^2-2 x +1 $
$ \left.\frac{ dy }{ dx }\right|_{ Q }=3 a ^2-2 a +1$
Hence $3 a^2-2 a+1=22$
$ \therefore 3 a^2-2 a-21=0 $
$ 3 a^2-9 a+7 a-21=0$

$ a=3$
$ b=21 |2 a+9 b|=195$
at $a =-7 / 3$ tangent will be parallel
Hence it is rejected