Determine the point on the curve y^2=4x that is closest to the point (1,0)
a) (0,0)
b) (1,2)
c) (0.5,sqrt of 2)
the distance d is found by
d^2 = (1-x)^2 + y^2
= 1-2x+x^2+4x
= (1+x)^2
d = 1+x
this is obviously a minimum when x=0, so (a)
To determine the point on the curve y^2 = 4x that is closest to the point (1,0), we need to find the minimum distance between the given point and any point on the curve.
Step 1: Calculate the distance between the point (1,0) and a general point (x, y) on the curve y^2 = 4x using the distance formula:
Distance = sqrt((x - 1)^2 + (y - 0)^2)
Step 2: Rewrite the equation of the curve in terms of y:
y^2 = 4x
Step 3: Substitute y^2 = 4x into the distance formula:
Distance = sqrt((x - 1)^2 + (4x - 0)^2)
Step 4: Simplify the distance formula:
Distance = sqrt((x - 1)^2 + 16x^2)
Step 5: Find the derivative of the distance formula with respect to x:
d(Distance)/dx = (1/2) * (2(x - 1) + 32x)
Simplifying the derivative:
d(Distance)/dx = x - 1 + 16x
Step 6: Set the derivative equal to zero and solve for x:
x - 1 + 16x = 0
17x - 1 = 0
17x = 1
x = 1/17
Step 7: Substitute the value of x back into the equation of the curve to find the corresponding y-coordinate:
y^2 = 4(1/17)
y^2 = 4/17
y = Âħsqrt(4/17)
Therefore, there are two possible points on the curve:
- (1/17, sqrt(4/17))
- (1/17, -sqrt(4/17))
Out of the given answer choices, the closest point to (1,0) is (0.5, sqrt(2)), which is closest to the point (1,0). Therefore, the correct answer is c) (0.5, sqrt(2)).
To determine the point on the curve y^2 = 4x that is closest to the point (1, 0), we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) = (1, 0). We need to find the value of y, let's call it y0, on the curve y^2 = 4x that minimizes the distance to (1, 0).
Substituting y^2 = 4x into the distance formula, we get:
Distance = sqrt((x2 - 1)^2 + (y^2 - 0)^2)
Now, we need to minimize this distance. Since y^2 = 4x, we can substitute 4x for y^2 in the equation:
Distance = sqrt((x2 - 1)^2 + (4x - 0)^2)
Taking the derivative of the distance formula with respect to x and setting it equal to zero will give us the x-coordinate of the point that minimizes the distance.
Let's calculate the derivative of the distance formula:
d(Distance)/dx = ((x2 - 1)(2) + (4x - 0)(8)) / (2 sqrt((x2 - 1)^2 + (4x - 0)^2))
Setting this derivative equal to zero and solving for x will give us the x-coordinate of the point that minimizes the distance.
0 = ((x2 - 1)(2) + (4x - 0)(8)) / (2 sqrt((x2 - 1)^2 + (4x - 0)^2))
Simplifying this equation will give us the value of x. To find the corresponding y-coordinate, we can substitute the value of x into the equation y^2 = 4x.
Let's solve the equation to find the value of x and then substitute it into y^2 = 4x to find the y-coordinate.