question of differentiation?

Question

verify tat,for all values of t,the point P ( at,at^3) lies on the curve y= x^3 / a^2 and find the equation of the tangent to the curve at this point.if the tangent at P meets the x-axis and y axis at Q and R respectively,find area of triangle OQR and the equation of locus midpoint of QR as P moves on original curve

sophist..can u continue?

yea..sophist..if can..can u do as much as u can

Answer 1

Starting at the beginning...

If the point P is defined by ( at,at^3), what this means is:

For any t (t=1, t=2, etc) you can calculate a point with x/y coordinates. The point is called P and it has coordinates (x,y), where x=a*t and y=a*t^3.

So here is the first step:

If x=a*t and y=a*t^3, can you find a way to express y in terms of x?

Let me know if you want more help.

(Okay, here goes)

Try to get y in terms of x. Y is expressed in terms of t^3, and x is expressed in terms of t^1. That's not very helpful. So try cubing x: x^3 = a^3 t^3.

Then you should be able to show that y=x^3 / a^2.

To find the tangent to a curve, you take its derivative. The derivative gives you the slope of the line. The point P will help you determine the full equation of the line in terms of "y=ax+b"

Draw a very good, very large, very detailed and accurate graph. Choose any point P on the line. Draw the tangent at that point; see where it intersects the x-axis and call that Q. See where it intersects the y-axis and call that R. Have all these points labeled clearly.

You say: "find the area of the triangle OQR." We don't know what O is. Is it possible you meant P?

Anyways, the rest is geometry. Just label the graph. Label the midpoint of the line QR. What can you say about it?

If you need more help, let me know what you've done and which parts confuse you.....

Answer 2

Interesting problem, but would require too much time to solve.