The-tangent-to-the-curve-y-ax-2-bx-2-at-1-1-2-is-parallel-to-the-normal-to-the-curve-y-x-2-6x-10-at-2-2-Find-the-values-of-a-and-b-

Question Number 171315 by pete last updated on 12/Jun/22

The tangent to the curve y = {ax}^{2} + bx + 2

at (1, \frac{1}{2}) is parallel to the normal to the curve

y = x^{2} + 6 x + 10 at (- 2, 2) . Find the values

of a and b .

Answered by som(math1967) last updated on 12/Jun/22

T a n g e n t t o t h e c u r v e y = {a x}^{2} + b x + 2

a t (1, \frac{1}{2}) ∴ \frac{1}{2} = a + b + 2

\Rightarrow a + b = - \frac{3}{2} \dots . . i)

\Rightarrow

s l o p e o f t a n g e n t

\frac{d y}{d x} = 2 a x + b

{[\frac{d y}{d x}]}_{1, \frac{1}{2}} = 2 a + b

s l o p e o f n o r m a l

y = x^{2} + 6 x + 10

- \frac{d x}{d y} = \frac{- 1}{2 \times - 2 + 6} = \frac{- 1}{2}

∴ 2 a + b = - \frac{1}{2} \dots \dots i i)

f r o m i) a n d i i)

a = - \frac{1}{2} + \frac{3}{2} = 1

b = - \frac{3}{2} - 1 = \frac{- 5}{2}

Commented by pete last updated on 12/Jun/22

thank you very much sir, i am grateful .

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