A point in the first quadrant lies on the parabola y = x^2. Express the coordinates of P as function of the angle of inclination of the line joining P to the origin
The angle joining P to the origin is
A = tan^-1 (y/x)
You can invert that to read
y = x tan A
In addition to that, along the parabola,
y = x^2
Therefore
x = tan A and
y = tan^2 A
To express the coordinates of point P on the parabola y = x^2 as a function of the angle of inclination of the line joining P to the origin, we can use trigonometry.
Let's assume the angle of inclination is θ. The line joining P to the origin can be represented by the equation y = mx, where m is the slope of the line. The slope, m, can be determined by taking the tangent of the angle θ.
Since the point P lies on the parabola y = x^2, the coordinates of P will be (x, x^2).
Now, let's find the slope of the line using trigonometry:
m = tan(θ)
Since the line passes through the point (x, x^2), we can write the equation of the line as:
y = tan(θ) * x
Since the y-coordinate of P is x^2, we can substitute this value into the equation:
x^2 = tan(θ) * x
Rearranging this equation, we get:
x^2 - tan(θ) * x = 0
This is a quadratic equation, which we can solve for x using the quadratic formula:
x = [ -b ± √(b^2 - 4ac) ] / 2a
For this equation, a = 1, b = -tan(θ), and c = 0.
Substituting these values into the quadratic formula, we get:
x = [ -(-tan(θ)) ± √((-tan(θ))^2 - 4 * 1 * 0) ] / (2 * 1)
Simplifying further:
x = [ tan(θ) ± √(tan^2(θ)) ] / 2
x = [ tan(θ) ± tan(θ) ] / 2
There are two potential values for x, one positive and one negative:
x₁ = (tan(θ) + tan(θ)) / 2 = tan(θ)
x₂ = (tan(θ) - tan(θ)) / 2 = 0
Since we are only considering the first quadrant, we can ignore the solution x₂ = 0.
Therefore, the coordinates of point P as a function of the angle of inclination θ are:
P = (tan(θ), (tan(θ))^2)
To express the coordinates of a point in the first quadrant lying on the parabola y = x^2 as a function of the angle of inclination of the line joining the point to the origin, we need to consider the trigonometric relationships between the coordinates and the angle.
Let's assume the point lying on the parabola is P(x, y), where x and y are the coordinates of the point. To find the angle of inclination of the line joining P to the origin (θ), we can use the tangent function, given by tan(θ) = y / x.
Rearranging this equation, we have y = x * tan(θ). Since the point P lies on the parabola y = x^2, we can substitute the value of y from the previous equation into the equation of the parabola:
x^2 = x * tan(θ)
This is a quadratic equation in terms of x. Solving it will give us the coordinates of the point P as a function of the angle of inclination (θ) of the line.
To solve the equation, we can rearrange it to the standard quadratic form:
x^2 - (tan(θ)) * x = 0
Next, we can factor out the common factor x:
x * (x - tan(θ)) = 0
Now, we have two possibilities:
1. x = 0
If x = 0, then y = 0 (since P lies on the parabola y = x^2), and the point P coincides with the origin (0, 0).
2. x - tan(θ) = 0
If x - tan(θ) = 0, then x = tan(θ). Substituting this value into y = x * tan(θ), we get y = (tan(θ))^2.
Therefore, the coordinates of the point P lying on the parabola y = x^2 as a function of the angle of inclination (θ) of the line are:
P(tan(θ), (tan(θ))^2)