A point P in the first quadrant lies on the parabola ...

Question

A point P in the first quadrant lies on the parabola 𝔂 = 𝔁². Express the coordinates of P as functions of the angle of inclination of the line joining P to the origin.

Accepted Answer

Welcome back, everyone. A pointr on the graph of the equation Y equals 8 x 2d lies in the first quadrant. If the line joining R to the origin makes an angle beta with a positive X axis, express the Y coordinate of R as a function of beta where given for answer chooses A Y equals C22 beta divided. 8 B says Y equals tent squad of beta divided by 8 C Y equals 2 beta divided by 8 and D Y equals cosine beta divided by 8. So let's visualize this. Problem using a coordinate plane. So we're given our function, which is a parabola, right? It is not really important to understand what type of a function we have for this problem, but we are going to use the exact function. So 8 x 2 is a parabola, and we're interested in the first quadrant. We're given some point R which belongs to the given function, the given parabola. And essentially what we're doing is connecting the origin to that specific point, and what we know is that the angle formed between that line and the positive x axis is beta, so we can essentially treat. The resultant The line has a hypotenuse of a right triangle where the adjacent side belongs to the x axis and the opposite side belongs to the y axis right or basically is parallel to the y axis that'd be more accurate to say so we can say that we have our Y, we have our X. And now we have to recall Soatoa. So it says that if we take the opposite side and divide by the adjacent side, such that we get Y divided by X, we're going to get agent of beta. Now what we can sell from here is that. If we essentially take. This expression and multiply both sides by X, then Y equals x multiplied by tangent of beta. From here we can also see that y is given to us, right? So if Y is given, we can just substitute it and on the left hand side we're going to get 8 x 2d because this is how we define Y and 8 x 2 is going to be equal to x multiplied by a tangent of beta. Now we can solve for X, and first of all, let's divide both sides by X. We can do that because X is not equal to 0. X and Y, those should be greater than 0 because we have the first quadrant, right? So we are allowed to divide by X, which gives us 8. X equals tangent of beta. So that x is equal to tangent of beta divided by 8. From here what we're going to do is simply substitute this expression into the original function Y equals 8 x 2. So this means that it is 8 multiplied by tangent of beta. Divided by 8 squared. Which gives us 8 multiplied by. 1/8 squared, right, we can take out 1/8. Multiplied by tangent squared of beta. And this gives us tangent squared of beta divided by 8, right? Because we have 8 divided by 8 squad, so we still have 8 in the denominator, and this essentially corresponds to the answer choice B. Which is our final answer. Thank you for watching.

A point P in the first quadrant lies on the parabola 𝔂 = 𝔁². Express the coordinates of P as functions of the angle of inclination of the line joining P to the origin.

Key Concepts

Parabola

Polar Coordinates

Angle of Inclination

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