Finding the Positive Number A in a Parabola Problem

In summary, the problem is to find the value of A for which the parabola y=x^2+3 has two tangents passing through the point (0,-2). These tangents are tangent to the parabola at (A,A^2+3) and (-A,A^2+3) and have a slope of 2A. Using the derivative of y=x^2, we can set the slope of the tangent lines equal to 2A and solve for A. The final answer for A can be determined by solving the equation A^2+3=3A^2-2.
  • #1
i3uddha
3
0
alright so here is the problem.
The parabola y=x**2+3 has two tangents which pass through the point (0,–2). One is tangent to the to the parabola at (A,A2+3) and the other at (–A,A2+3). Find (the positive number) A.

i've attempted this problem several times but haven't been able to find the solution. :confused:
could someone tell me step by step how i would start this problem?
 
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  • #2
Better to use x^2 rather than x**2 (I haven't seen that notation since BASIC). Any line through (0, -2) can be written y= mx- 2. A line tangent to y= x^2+ 3 at (A, A^2+ 3) must have slope m= 2A: y= 2Ax- 2
Determine A so that y= A^2+ 3= 2A(A)- 2= 3A^2- 2.
 
  • #3
how did you get the slope to be 2A?
and what would the answer for A be?
 
  • #4
The derivative of y= x^2 is y'= 2x. At x= A, y'= 2A and that is the slope ofthe tangent line.

Surely you can solve A^2+ 3= 3A^2- 2 yourself.
 

FAQ: Finding the Positive Number A in a Parabola Problem

How do I determine the positive number A in a parabola problem?

To find the positive number A in a parabola problem, you can use the formula A = 1/(4p) where p is the distance from the vertex to the focus. Alternatively, you can use the equation A = 4a where a is the distance from the vertex to the directrix.

What is the significance of finding the positive number A in a parabola problem?

The positive number A in a parabola problem is the coefficient of the squared term in the parabola's equation. It determines the shape and orientation of the parabola. A positive A value indicates an upward opening parabola, while a negative A value indicates a downward opening parabola.

Can I find the positive number A by looking at the graph of a parabola?

Yes, you can estimate the positive number A by looking at the graph of a parabola. The value of A can be determined by the steepness of the parabola's curve. A steep curve indicates a larger A value, while a flatter curve indicates a smaller A value.

How does changing the positive number A affect the parabola?

Changing the positive number A affects the parabola by changing its shape and orientation. A larger A value results in a narrower and steeper parabola, while a smaller A value results in a wider and flatter parabola. Additionally, changing A also affects the vertex, focus, and directrix of the parabola.

Are there any real-life applications of finding the positive number A in a parabola problem?

Yes, there are many real-life applications of finding the positive number A in a parabola problem. For example, parabolic mirrors and satellite dishes use the properties of parabolas to focus light and signals. The shape and orientation of the parabola, determined by the positive number A, are crucial in these applications.

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