Consider the curves `y=x^(2)+2` and `y=10-x^(2)` . Let `theta` be the angle between both the curves at point of intersection, then find `|tan theta|` (a) `(8)/(15)` (b) `(5)/(17)` (c) `(3)/(17)` (d) `(8)/(17)`

Options

A. `(7)/(17)`
B. `(8)/(15)`
C. `(4)/(9)`
D. `(8)/(17)`

Answer

B

Explanation

Given equation of curves are
`" " y = 10 - x ^(2)" " `…(i)
and `" " y = 2 + x^(2)" " ` … (ii)
For point of intersection, consider
`" " 10- x^(2) = 2 + x ^(2)`
`rArr " " 2x^(2) = 8`
`rArr " "x ^(2) = 4`
`rArr " " x= pm 2 `
Clearly, when `x =2` , then `y =6` (using Eq. (i)) and when `x = - 2` , then `y = 6`
Thus, the point of intersection are `(2, 6) and (-2, 6)`.
Let `m_1` be the slope of tangent to the curve (i) and `m_2` be the slope of tangent to the curve (ii)
For curve (i) `(dy)/(dx) = -2 x and ` for curve (ii) `(dy)/(dx)= 2x`
`therefore ` At ` (2, 6)`, slope `m_1 =-4 and m_2=-4` and in that case
`|tan theta| = |(m_2 - m_1)/(1+ m_1m_2)| = | (-4 -4)/( 1-16)| = (8)/(15)`
At `(- 2, 6)`, slopes `m_1 = 4 and m_2 = - 4 ` and in that case
`|tan theta| = |(m_2 - m_1)/(1+m_1m_2)| = |(-4-4)/(1-16)|= (8)/(15)`