if the line y=1.732(x-1.732) cuts the parabola y^2=x+2 at the point A and B. find PA.PB, where P is the point where line cuts the x-axis.
looks like your straight line equation was
y = √3(x - √3) or y = √3x - 3
intersect that with y^2 = x+2
from the straight line: x = (y+3)/√3
y^2 = (y+3)/√3 + 2
√3y^2 = y + 3 + 2√3
√3y^2 - y - (3+2√3) = 0
y = (1 ± √(1 - 4√3(-3-2√3 )/(2√3)
= 2.242 or -1.665 (you had 3 decimal places)
At P, y = 0
x = 1.732 (looks like √3 exactly)
AP = √(2.242-0)^2 + (3.026-1.732)^2 ) = 2.589
PB = √(-1.665-0)^2 + (1.732-.771)^2 ) = 1.922
PA.PB = 4.976
You better check my arithmetic on that one.
To find the solution, we need to solve two equations: the equation of the parabola and the equation of the line, in order to find the coordinates of point P.
Let's start by equating the equations of the line and the parabola:
1.732(x - 1.732) = ±√(x + 2)
Now, let's square both sides of the equation in order to remove the square root:
3(x - 1.732)^2 = x + 2
Expand the expression:
3(x^2 - 3.464x + 3) = x + 2
Distribute the 3:
3x^2 - 10.392x + 9 = x + 2
Rearrange the equation:
3x^2 - 11.392x + 7 = 0
Now, we have a quadratic equation. We can solve this equation by factoring, completing the square, or by using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation:
a = 3, b = -11.392, c = 7
x = (-(-11.392) ± √((-11.392)^2 - 4 * 3 * 7)) / (2 * 3)
Simplifying:
x = (11.392 ± √(130.057344 - 84)) / 6
x = (11.392 ± √46.057344) / 6
x = (11.392 ± 6.7812) / 6
We get two possible x-coordinates for point P:
x₁ = (11.392 + 6.7812) / 6 ≈ 3.62
x₂ = (11.392 - 6.7812) / 6 ≈ 0.435
Now, substitute these values back into the equation of the line y = 1.732(x - 1.732) to find the corresponding y-coordinates.
For x₁ = 3.62:
y₁ = 1.732(3.62 - 1.732) ≈ 4.298
For x₂ = 0.435:
y₂ = 1.732(0.435 - 1.732) ≈-1.432
So, point P will have coordinates (3.62, 4.298) and (0.435, -1.432).
Now, let's find points A and B:
Substitute the coordinates of point P (3.62, 4.298) into the equation of the parabola y^2 = x + 2:
(4.298)^2 = 3.62 + 2
18.45 = 5.62
However, this equation does not hold true, which means that the line y = 1.732(x - 1.732) does not intersect the parabola at point A.
Therefore, we do not have point A.
Now, let's find point B:
Substitute the coordinates of point P (0.435, -1.432) into the equation of the parabola y^2 = x + 2:
(-1.432)^2 = 0.435 + 2
2.057024 = 2.435
Again, this equation does not hold true, indicating that the line does not intersect the parabola at point B either.
Therefore, we do not have point B.
As a result, we cannot calculate the product PA.PB since points A and B do not exist in this scenario.