Sum of slopes of common tangent to `y = (x^(2))/(4) - 3x +10` and `y = 2 - (x^(2))/(4)` is (a) -6 (b) -3 (c) `1/2` (d) none of these

Options

A. `-6`
B. `-3`
C. `1//2`
D. none of these

Answer

B

Explanation

Let `y = mx +c` be a common tangent to
`y = (x^(2))/(4) -3x +10` (i)
and `y =2 -(x^(2))/(4)` (ii)
On solving `y = mx +c` with (i), we get
`mx +c = (x^(2))/(4) -3x +10`
or `(x^(2))/(4) -(m+3) x +10 -c =0`
`D =0` gives `(m+3)^(2) =10 -c`
`rArr c = 10 -(m+3)^(2)` (iii)
Similarly solving line with (ii), we get
`mx +c =2 -(x^(2))/(4)`
or `(x^(2))/(4) + mx +c-2 =0`
`D =0` gives `m^(2) = c-2`
`rArr c =2 +m^(2)` (iv)
From (i) and (iv), we get
`10 -(m+3)^(2) =2 +m^(2)`
`rArr 2m^(2) +6m +1 =0`
`rArr m_(1) + m_(2) =- (6)/(2) =- 3`