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The tangent to the curve y=x^(2)-5x+5, p...

The tangent to the curve `y=x^(2)-5x+5`, parallel to the line 2y=4x+1, also passes through the point

A

`((1)/(4), (7)/(2))`

B

`((7)/(2), (1)/(4))`

C

`(-(1)/(8), 7)`

D

`((1)/(8), -7)`

Text Solution

Verified by Experts

The correct Answer is:
D
The given curve is ` y = x ^(2) - 5x + 5 " " ` … (i)
Now, slope of tangent at any point `(x, y)` on the curve is
`" " (dy)/(dx) = 2x - 5" "`… (ii)
`" " `[ on differentiating Eq. (i) w.r.t. x]
`because` It is given that tangent is parallel to line
` " " 2y = 4x + 1 `
So, ` (dy)/(dx) = 2 [ because` slope of line ` 2y = 4x + 1 ` is 2`]`
`rArr 2x - 5 = 2 rArr 2x = 7 rArr x = (7)/(2)`
On putting `x= (7)/(2)` in Eq. (i), we get
`" " y = (49)/( 4) - ( 35)/( 2) + 5 = ( 69)/( 4) -( 35)/( 2) = - (1)/(4)`
Now, equation of tanent to the curve (i) at point `((7)/(2), - (1)/(4))` and having slope 2 , is
`" " y + (1)/(4) = 2(x -(7)/(2)) rArr y + (1)/(4) = 2x - 7`
`rArr " " y = 2x - (29)/(4)" "` ...(iii)
On checking all the options, we get the point `((1)/(8), -7)` satisfy the line (iii).
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