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the tangents on the ellipse (x^2)/9 + (y^2)/4 = 1 are drawn from the point (-3,5). What...

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nina-xyz | Student, Grade 11 | eNoter

Posted April 3, 2013 at 10:05 PM via web

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the tangents on the ellipse (x^2)/9 + (y^2)/4 = 1 are drawn from the point (-3,5). What is the angle they close?

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oldnick | Valedictorian

Posted April 4, 2013 at 2:02 AM (Answer #1)

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First of all we note that the ellipse touches the x-axis at the points:  x = - 3  e x = 3

                                                  
explicit respect to x we get:  y =2/3 sqrt ( 9 - x^2)                 

The slope is:    y’= .  -   2x     .
                          3 sqrt(9 – x2).

 

A line tangent to the already know, as the abscissa of the point P is one in which the ellipse touches the x axis, then it will be vertical equation x = -3 then the point of tangency will be the point

Q (-3.0) . We have to find the other point R of tangency.

 

To do this just put the system of equations, one stored at the point R on the ellipse and another of tangency to the same:


   x2 + y2 = 1
   9     4

 (y – y0) = m ( x – x0)

 

                 .         .

Thus:  [ 2 sqrt(9 – x2)   –  5 ] =  .  -2x  . ( x + 3)
            3                                   3 sqrt(9 – x2)

 

 

that once developed, it leads to:


29x2 – 24x -189 = 0  

 

 

of solutions: x = - 3 (as we foretold), and x = 63/29 (about 2,17)  which   R( 63 ; 40)
               29

                                                                                                            
Therefore, the tangent  of the angle a we’r searching for  is: 

 

tg a =  ( 40 -5) /  (63/29  + 3) = 0,25  

 

corresponding to the angle:  14° 2’10s .      
 

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pramodpandey | College Teacher | Valedictorian

Posted April 4, 2013 at 6:02 AM (Answer #2)

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Let equation of the tangents from point (-3,5) be

y=mx+c                       (i)

(i) posses through (-3,5) so,

5=-3m+c

c=3m+5                       (ii)

Line (i) will be tangent to ellipse `x^2/9+y^2/4=1`    if

`c=+-sqrt(9m^2+4)`         (iii)

from (ii) and (iii) ,we have

`(3m+5)^2=9m^2+4`

`m=-7/10`

`c=2.9`

y=`-.7x+-2.9`             (iv)

let angle between lines of (iv) be `theta`

`tan(theta)=(-.7-(-.7))/(1+(-.7)(-.7))`

   `tan(theta)=0`

`theta=0`

lines (iv)  are parallel.

Thus distance between them= `2xx2.9`

`=5.8` unit

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