if the line y=2x+b and the parabola y^2=8(x+2) meet exactly at one point,then what is the value of b
Non-Calculus way:
Let's find their intersection, ...
(2x+b)^2 = 8(x+2)
4x^2 + 4bx + b^2 - 8x - 16 = 0
4x^2 + (4b-8)x + b^2 - 16 = 0
this is a quadratic with
A = 4
B = 4b -8
C = b^2 - 16
IF this is to have only one root, then the B^2 - 4AC, or the discriminant, must be zero.
so
(4b-8)^2 - 4(4)(b^2 - 16) = 0
16b^2 - 64b + 64 - 16b^2 + 256 = 0
64b = 320
b =5
Using Calculus:
for y^2= 8x + 16
2y(dy/dx) = 8
dy/dx = 8/2y ----> slope of the tangent
but we know the slope of the tangent is 2 from y - 2x+b
then 8/2y = 2
4y = 8, then y = 2
sub that into parabola
4 = 8x + 16
x = -12/8 =- 3/2
so the point of contact is (- 3/2,2) which must satisfy
y = 2x + b
2 = 2(-3/2) + b
2 = -3 + b
b = 5 , just like before.
To find the value of b, we need to determine the point of intersection between the line y = 2x + b and the parabola y^2 = 8(x + 2).
First, let's substitute the equation of the line into the equation of the parabola to solve for x:
(2x + b)^2 = 8(x + 2)
Simplifying the equation:
4x^2 + 4bx + b^2 = 8x + 16
Rearranging this equation:
4x^2 + (4b - 8)x + (b^2 - 16) = 0
For the line and parabola to intersect at exactly one point, the discriminant of this quadratic equation should be zero.
The discriminant can be calculated using the formula b^2 - 4ac, where a = 4, b = 4b - 8, and c = b^2 - 16:
(4b - 8)^2 - 4(4)(b^2 - 16) = 0
Expanding and simplifying the equation:
16b^2 - 64b + 64 - 16b^2 + 256 = 0
Combining like terms:
-48b + 320 = 0
Solving for b:
-48b = -320
b = -320 / -48
b = 40/6 or 20/3
Therefore, the value of b is 20/3 or approximately 6.67.
To find the value of b in the given scenario, we need to find the point of intersection between the line y = 2x + b and the parabola y^2 = 8(x + 2).
Let's start by substituting the value of y from the line equation into the parabola equation:
(2x + b)^2 = 8(x + 2)
Expanding the left side:
4x^2 + 4bx + b^2 = 8x + 16
Rearranging the terms:
4x^2 + (4b - 8)x + (b^2 - 16) = 0
For the given scenario, the line and parabola intersect at exactly one point. This implies that the quadratic equation has only one root, which means the discriminant (b^2 - 4ac) should be equal to zero, where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0.
So, in our equation, the discriminant is:
(4b - 8)^2 - 4(4)(b^2 - 16) = 0
Expanding and simplifying:
16b^2 - 64b + 64 - 16b^2 + 256 = 0
Combining like terms:
-48b + 320 = 0
Rearranging and solving for b:
48b = 320
b = 320/48
b = 6.67 (rounded to two decimal places)
Therefore, the value of b in the given scenario is approximately 6.67.