Find the point of intersection of the tangents to the curve y = x^2 at the points (-1/2, 1/4) and (1, 1).
- print Print
- list Cite
Expert Answers
calendarEducator since 2008
write3,662 answers
starTop subjects are Math, Science, and Social Sciences
y= x^2
The slope of tangent line of the curve y is the first derivative at the point ( -1/2, 1/4).
==> y= 2x.
==> m = 2*-1/2 = -1
==> (y-1/4) = m ( x+1/2)
==> y-1/4 = -1(x+1/2)
==> y= -x - 1/2 + 1/4
==> y= -x -1/4
==> 4y + 4x + 1 = 0
Now we will find the tangent line at the point (1,1).
==< Then, the slope is m = 2*1 = 2
==> (y-1) = 2(x-1)
==> y= 2x - 2 + 1
==> y= 2x -1
Now we will determine the intersection points between both lines.
==> y= 2x - 1
==> y = -x...
(The entire section contains 2 answers and 241 words.)
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.
Related Questions
- Find an equation of the tangent line to the given curve at the specified point.y=...
- 4 Educator Answers
- Find the slope of the tangent line to the curve (a lemniscate) `2(x^2+y^2)^2 = 25(x^2-y^2)` at...
- 1 Educator Answer
- `y = ln(x^2 - 3x + 1), (3,0)` Find an equation of the tangent line to the curve at the...
- 1 Educator Answer
- `y = sqrt(1 + x^3), (2, 3)` Find an equation of the tangent line to the curve at the given...
- 1 Educator Answer
- find the exact length of the curve, `x=(1/8)y^4+1/(4y^2)` `1<=y<=2` please explain as...
- 1 Educator Answer
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
To find the intersection of tangents to y = x^2.
The equation of tangent to a curve at (x1,y1) is given by:
y-y1 = (dy/dx)(x-x1).
So dy/dx = (x^2)' = 2x.
At x= (-1/2), dy/dx = 2(-1/2) = -1.
So the tangent at (-1/2, 1/4) is given by:
y-1/4 = (-1)(x+1/2). Or
y-1/4 = -x-1/2.
y = -x-1/2+1/4 = -x-1/4.
y = -x-1/4...................(1).
Similarly at x = 1, dy/dx = 2x = 2*1 = 2. So the tangent at (1,1) is given by:
y-1 = 2(x-1)
=>y = 2x-2+1 = 2x-1. Or
=>y = 2x-1...........(2).
y = -x-1/4.........(1)
From (1) and (2) we get:
2x-1 = -x-1/4
=>2x+x = -1/4 +1 = 3/4.
=>3x= 3/4.
=> x = (3/4)/3 = 1/4.
Put x= 1/4 in eq (1) and we get y = -x-1/4 = -1/4-1/4 = -1/2.
Therefore the point of intersection of tangents is at (1/4, -1/2).
Student Answers