Find the point on the curve `y=x^(2)+1` at which the tangent drawn makes an angle of `45^(@)` from X-axis.

Options

A. `(1/2,5/4)`.
B. `(1/2,3/4)`.
C. `(-1/2,5/4)`.
D. None of these

Answer

A

Explanation

`y=x^(2)+1`
`rArr (dy)/(dx) = 2x`
Let the point be `(x_(1),y_(1))`.
Slope of tangent at point `(x_(1), y_(1))`
`m= 2x_(1)`
But given that m = tan `45^(@)` = 1
`:gt 2x_(1) = 1`
`rArr x_(1) = 1/2`
Now, the point `(x_(1),y_(1))` lies on the curve ` y = x^(2) +1`.
`:. y_(1)=x_(1)^(2)+1`
`=(1/2)^(2)+1 =5/4`
Therefore the required point = `(1/2,5/4)`.