Graphing to find the intersections of lines and a parabola in this limit

[Jalal_khan]

Homework Statement

I am doing this for fun.

Relevant Equations

Linear equations, cubic models and etc

Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question.

Question:

Let P(k,k^2) be a point on the parabola y=x^2 with k>0.
Let O denote the origin.
Let A(0, a)denote the y-intercept of the perpendicular bisector of the segment OP.

What is the limit of a as P approaches O?

Thank you!

 

[andrewkirk]

You need to write a formula that expresses the y intercept Y (distance AO) in terms of k. Then the answer will be the limit of that formula as $k\to 0$.

To write a formula, you'll need to do a bit of geometry and trigonometry. Draw the point X where the bisector AM hits the x axis. Label as $\theta$ the angle $\angle MOX$, and write as many other angles in the diagram as you can in terms of $\theta$. Label as many line segments as you can in terms of $k$.
Then try to use trigonometry to express $\theta$ in terms of $k$.

If you haven't learned trig, you can instead try using Pythagoras's theorem. It might take a little longer, but you'd get there.

From there you should be able to express AO in terms of $\theta$ and $k$, and hence in terms of $k$ alone.

Then take the limit and you're finished!

 

[SammyS]

Homework Statement:: I am doing this for fun.
Relevant Equations:: Linear equations, cubic models and etc

Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question.

Question:

Let P(k,k^2) be a point on the parabola y=x^2 with k>0.
Let O denote the origin.
Let A(0, a)denote the y-intercept of the perpendicular bisector of the segment OP.

What is the limit of a as P approaches O?

Thank you!

Hello @Jalal_khan .

Here at PF (Physics Forums), you need to show an attempt at a solution. In your case, I suppose that the image you provided will do.

If you are trying to do this problem purely by drawing a graph using pencil, paper and a ruler, you need to use the same scale factor for both axes, otherwise it's virtually impossible to draw perpendicular lines. That's also true if you are using a graphing program/app.

Otherwise:
You don't need to use trigonometry. You can use algebra and knowledge of slope of a line and how slopes are related for pairs of perpendicular lines.