My First Stanford Math Tournament Problem

TL;DR

Can the curve y=x^2 be rotated to pass through the point (5,0)? Yes, and the angle of rotation can be found using trigonometry.

Transcript

okay as we all know the curve y=x^2 this right here does not pass this   point five comma zero but is it possible for us  to rotate the graph so that it does pass five comma zero   yes of course because we can see it from the  pictures right here right okay i know the main   question should be by what angle now we have  to rotate so that this right... Read More

Key Insights

• 💁 The cartesian equation of a parabola can be transformed into polar form using the variables rsin(theta) and rcos(theta).
• 🪜 The rotated curve can be found by adding the rotation angle to the original equation.
• 🥡 The angle of rotation can be determined by taking the inverse tangent of x squared divided by x in the intersection triangle.

Questions & Answers

Q: How is the cartesian equation of y=x^2 transformed into its polar form?

To convert the cartesian equation y=x^2 into polar form, the variables x and y are replaced with rsin(theta) and rcos(theta) respectively.

Q: How is the angle of rotation determined?

The angle of rotation is found by using the inverse tangent function on the ratio of x squared to x, which gives the angle theta.

Q: How is the intersection point of the curve and circle found?

By substituting x squared into the quadratic equation x^2 + x^2 - 25 = 0, the solutions for x are obtained. The positive solution represents the intersection point.

Q: How can the equation of the rotated parabola be written?

The equation of the rotated parabola can be written in polar form as r = sqrt(-1 + sqrt(101))/2.

Summary & Key Takeaways

• The video discusses rotating the curve y=x^2 to make it pass through the point (5,0).

• The cartesian equation is transformed into its polar form.

• The angle of rotation is determined by finding the intersection point of the curve and a circle with radius 5.