The value of line MN is 9
The point where the tangent touches the curve is the point of tangency. Lines or segments can create a point of tangency with a circle or a curve. Two circles can also have a common point of tangency if they touch, but do not intersect.
Since, we have that the tangents are of equal lengths, then, we get that;
8 + 10 = 2x - 7 + 9
collect the like terms, we get;
18 = 2x -7 + 9
9 = 2x - 7
2x = 9 + 7
add the values
2x = 16
Divide both sides by the coefficient
x = 8
Then, line MN = 2x - 7
substitute the value
MN = 2(8) - 7
MN = 9