A point P in the first quadrant lies on the parabola y=x^2. Express the coordinates of P as functions of the angle of inclination of the line joining p to the origin.
I honestly don't think my math teacher explained this at all. Please help. Thank you very much.
To find the coordinates of point P in terms of the angle of inclination of the line joining P to the origin, we can use trigonometry.
Let's assume the angle of inclination of the line is θ. We can see that the line is formed by the tangent of θ.
Now, let's consider the point P on the parabola y = x^2. To express the coordinates of P as functions of the angle of inclination, we need to find the value of x and y in terms of θ.
Since P lies on the parabola, the coordinates of P can be written as (x, x^2).
Now, let's consider the right-angled triangle formed by the line joining P to the origin and the x-axis. The length of the opposite side to the angle θ is y = x^2, and the length of the adjacent side is x.
Using the definition of tangent, we have:
tan(θ) = opposite / adjacent
= (x^2) / x
= x.
So, we can conclude that x = tan(θ).
To find the value of y in terms of θ, we can substitute the value of x into the equation y = x^2:
y = (tan(θ))^2.
Therefore, the coordinates of point P can be expressed as functions of the angle of inclination (θ) as:
x = tan(θ),
y = (tan(θ))^2.
Hence, the coordinates of P are (tan(θ), (tan(θ))^2) in terms of the angle of inclination θ.
Pick a point, x,y
P(x,y)=(x,2x^2)
now, tanTheta=2x^2/x=2x
and the distance is the length of the hyp, or r= x/cosTheta
P(r,Theta)=( x/cosTheta, arctan 2x) in polar coordinates.