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Find the area of the triangle formed by the positive x-axis, and both the normal and the tangent to the circle x^2 + y^2 = 4 at (x,y) = (1,sqrt(3)).

 Feb 20, 2021
 #1
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Slope  of   any  tangent  line to   the   circle = -x/y

 

At  (1.sqrt 3)    the equation of the  tangent  line  is

 

y =  (-1/sqrt (3) ) ( x  -1)  + sqrt (3)

 

This  line  intersects  the  x axis  at

 

-sqrt (3)=  (-1/sqrt (3)(  x   -1)                     mult  through  by  sqrt (3)

 

3  =   (x  -1)

 

4 =   x  

 

The  equation  of  the normal  line  at  ( 1.sqrt 3)   is

 

y =  sqrt (3)  ( x  -1) + sqrt (3) 

 

y  =  sqrt (3) x   

 

This line intersects  the  x axis  at  x =   0

 

The  base of  the triangle  formed =   4

The height  of the triangle formed  =  sqrt (3)

 

Area = (1/2) (4)  sqrt (3)    =   2  sqrt (3)

 

Here's  a pic :

 

 

cool cool cool

 Feb 20, 2021

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