# Calc

A line tangent to y=(x^2)+1 at x=a, a>0, intersects the x-axis at point P.

a) Write an expression for the area of the triangle formed by the tangent line, the x-axis, and the line x=a.

b) For what value of 'a' is the area of the triangle a minimum? Justify your answer.

Show all work. Thanks!

1. 👍 0
2. 👎 0
3. 👁 98
1. quite a question for this forum.

I worked it out, but will only give you some of the steps, leaving it up to you to fill in most of the steps.

1. The graph is a parabola with vertex at (0,1)

2. Let A(a,a^2 + 1) be the point of contact

3. the slope of the tangent is the derivative of the function which is y' = 2x
At the point A the slope is then 2a

3. find the equation of the tangent line with slope 2a and point (a,a^2+1)

let y = 0 and solve for x to get the x-intercept
(you should get x = (a^2 - 1)/(2a)

call that point P
let the other point of the triangle which is on the x-axis be Q
Of course Q is (a,0)

So the base of the triangle is PQ and the height is AQ

we need PQ which is a - (x-intercept)
= a - (a^2 - 1)/(2a)

you should get (a^2+1)/(2a), this is your base of the triangle
the height AQ of course is a^2+1 from the equation.

Now Area = (1/2)base x height

Differentiate that, set it equal to zero and solve for a

(I got a = 1/√3)

1. 👍 0
2. 👎 0
posted by Reiny

## Similar Questions

1. ### Calculus

Consider line segments which are tangent to a point on the right half (x>0) of the curve y = x^2+7 and connect the tangent point to the x-axis. If the tangent point is close to the y-axis, the line segment is long. If the tangent

asked by Lynnn on April 8, 2012
2. ### Calc.

Find the area of the region bounded by the parabola y=x^2, the tangent line to this parabola at (1,1) and the x-axis. I don't really get what this question is asking. It looks like the area of right triangle to me...try the graph,

asked by Hebe on September 26, 2006
3. ### calculus

Consider line segments which are tangent to a point on the right half (x>0) of the curve y = x^2 + 1 and connect the tangent point to the x-axis. If the tangent point is close to the y-axis, the line segment is long. If the

asked by Hal on November 13, 2009
4. ### Calculus AB

Sorry but I've got a lot of problems that I don't understand. 1) Let f(x)= (3x-1)e^x. For which value of x is the slope of the tangent line to f positive? Negative? Zero? 2) Find an equation of the tangent line to the oven curve

asked by Annie on October 30, 2016
5. ### Math

Suppose p and q are points on the unit circle such that the line through p and q intersects the real axis. Show if z is the point where this line intersects the real axis, then z=(p+q)/(pq+1) So far I've figured out that the

asked by Katie on May 31, 2016
6. ### Precalculus

The figure below shows a conchoid of Nicomedes. The top part is formed by a ray from the origin that rotates through an angle t° with the x-axis. The ray intersects the fixed line y = 2 at point A. From A you measure out 7 more

asked by Anon on January 20, 2012
7. ### Math (Pre Cal)

Hi, I am studying for a precalculus quiz and I do not understand this hw problem: "The tangent line to a circle may be defined as the point that intersects a circle in a single point... If the equation of the circle is x^2+y^2=r^2

asked by Dealie on August 28, 2010
8. ### calculus

Hello there, assistance would be terrific, thank you very much. Consider the function f(x)= x^3 + 2x^2 + bx. a) The equation of the tangent line to the graph of this function at x = 1 is given by y = ? b) The tangent line

asked by Kevin Land on November 7, 2016
9. ### calc

standard equations of circles that have centers on line 5x-2y=-21 and are tangent to both axis. Any ideas?? First express the line as in point slope form y=(5/2)x - 21/2 The center must be on the line, so it is a point expressed

asked by Melba on September 26, 2006
10. ### maths

2. The tangent to the curve y = ex2 drawn at the point (e, ee) intersects the line joining the points (e - 1, ee-1) and (e+1, ee+1) 3. A hyperbola, having the transverse axis of length 4 sin, is confocal with the ellipse 3x2 + 4y2

asked by shailja on June 25, 2011

More Similar Questions