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If $\theta$ denotes the acute angle between the curves, $y = 10-x^2$ and $y = 2 +x^2$ at a point of their intersection, then $|\tan\theta|$ is equal to:
Option 1)
$\frac{4}{9}$
Option 2)
$\frac{8}{15}$
Option 3)
$\frac{7}{17}$
Option 4)
$\frac{8}{17}$

Answers (1)

best_answer

Angle of intersection of two curves -

The angle of intersection of two curves is the angle subtended between the tangents at their point of intersection.Let m₁ & m₂ are two slope of tangents at intersection point of two curves then

$tan\theta=\frac{[m_{1}-m_{2}]}{1+m_{1}m_{2}}$

- wherein

where $\theta$ is angle between two curves tangents.

Given curves are

$y=10-x^{2}$ and $y=2+x^{2}$

First find point of intersection

$P(2,6)$

Also $m_{_{1}}=\frac{\mathrm{d} y}{\mathrm{d} x}|_{P(2,6)}=-2x=-4$

and $m_{_{2}}=\frac{\mathrm{d} y}{\mathrm{d} x}|_{P(2,6)}=2x=4$

$\therefore \left | \tan \theta \right |=\left | \frac{m_{1}-m_{2}}{1+m_{1}m_{2}} \right |=\left | \frac{-4-4}{1+(-4)(4)} \right |=\frac{8}{15}$

Option 1)

$\frac{4}{9}$

Option 2)
$\frac{8}{15}$
Option 3)

$\frac{7}{17}$

Option 4)

$\frac{8}{17}$

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