The curves y=x²-1, y=8 x-x²-9

Correct Answer is : touch each other at (2,3)(2,3)(2,3)

intersect at right angles at (2,3)

Correct Answer is : touch each other at (2,3)(2,3)(2,3)

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AP EAPCET 07-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 66 of 160

Marks: +1, -0

The curves

y=x^{2}-1, y=8 x-x^{2}-9

Solution: 👈: Video Solution

y=x^2-1

\Rightarrow y=8 x-x^{2}-9

For point of intersection,

x^{2}-1=8 x-x^{2}-9

2 x^{2}-8 x+8=0

\Rightarrow x^{2}-4 x+4=0 .

\Rightarrow (x-2)^{2}=0

\Rightarrow x =2

\therefore y=2^{2}-1=3

Point of intersection

=(2,3)

\left[ \frac{dy}{dx} \right]_{c_1}=2 x

and

\left[ \frac{dy}{dx} \right]_{c_2} = 8-2x

\text{At } (2,3)

\left[ \frac{dy}{dx} \right]_{c_1} =4

and

\left[ \frac{dy}{dx} \right]_{c_2}=4

\therefore \tan \theta = \left| \frac{ \left[ \frac{dy}{dx} \right]_{c_1} - \left[ \frac{dy}{dx} \right]_{c_2} }{ 1+ \left[ \frac{dy}{dx} \right]_{c_1} \left[ \frac{dy}{dx} \right]_{c_2} } \right|

\dots \left| \theta \rightarrow \text{Angle between both curves} \right|

=0

Hence, both curves tough each other at

(2,3)

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