Answer:
D
Step-by-step explanation:
x/15 = tan 30= 1/√3
x = 15/√3
Answer:
![\rm \displaystyle y _{ \rm tangent} = - \frac{8}{5} x - \frac{5}{2} a](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20_%7B%20%5Crm%20tangent%7D%20%3D%20%20%20-%20%5Cfrac%7B8%7D%7B5%7D%20x%20-%20%20%5Cfrac%7B5%7D%7B2%7D%20a)
![\rm \displaystyle y _{ \rm normal} = \frac{5}{8} x - \frac{765}{128} a](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20_%7B%20%5Crm%20normal%7D%20%3D%20%20%5Cfrac%7B5%7D%7B8%7D%20x%20%20-%20%20%5Cfrac%7B765%7D%7B128%7D%20a)
Step-by-step explanation:
we are given a equation of parabola and we want to find the equation of tangent and normal lines of the Parabola
<u>finding</u><u> the</u><u> </u><u>tangent</u><u> </u><u>line</u>
equation of a line given by:
![\displaystyle y = mx + b](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20y%20%3D%20mx%20%2B%20b)
where:
- m is the slope
- b is the y-intercept
to find m take derivative In both sides of the equation of parabola
![\displaystyle \frac{d}{dx} {y}^{2} = \frac{d}{dx} 16ax](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%5Cfrac%7Bd%7D%7Bdx%7D%20%7By%7D%5E%7B2%7D%20%3D%20%20%5Cfrac%7Bd%7D%7Bdx%7D%2016ax%20)
![\displaystyle 2y\frac{dy}{dx}= 16a](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%202y%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20%2016a)
divide both sides by 2y:
![\displaystyle \frac{dy}{dx}= \frac{16a}{2y}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20%20%20%5Cfrac%7B16a%7D%7B2y%7D)
substitute the given value of y:
![\displaystyle \frac{dy}{dx}= \frac{16a}{2( - 5a)}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20%20%20%5Cfrac%7B16a%7D%7B2%28%20-%205a%29%7D)
simplify:
![\displaystyle \frac{dy}{dx}= - \frac{8}{5}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20%20%20%20-%20%5Cfrac%7B8%7D%7B5%7D)
therefore
![\displaystyle m_{ \rm tangent} = - \frac{8}{5}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%20m_%7B%20%5Crm%20tangent%7D%20%3D%20%20%20%20-%20%5Cfrac%7B8%7D%7B5%7D)
now we need to figure out the x coordinate to do so we can use the Parabola equation
![\displaystyle ( - 5a {)}^{2} = 16ax](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%28%20-%205a%20%7B%29%7D%5E%7B2%7D%20%20%3D%2016ax%20)
simplify:
![\displaystyle x = \frac{25}{16} a](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20x%20%3D%20%20%5Cfrac%7B25%7D%7B16%7D%20a)
we'll use point-slope form of linear equation to get the equation and to get so substitute what we got
![\rm \displaystyle y - ( - 5a)= - \frac{8}{5} (x - \frac{25}{16} a)](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20-%20%28%20%20-%205a%29%3D%20%20%20-%20%5Cfrac%7B8%7D%7B5%7D%20%28x%20-%20%20%5Cfrac%7B25%7D%7B16%7D%20a%29)
simplify which yields:
![\rm \displaystyle y = - \frac{8}{5} x - \frac{5}{2} a](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20%3D%20%20%20-%20%5Cfrac%7B8%7D%7B5%7D%20x%20-%20%5Cfrac%7B5%7D%7B2%7D%20a)
<u>finding</u><u> the</u><u> </u><u>equation</u><u> </u><u>of </u><u>the</u><u> </u><u>normal</u><u> </u><u>line</u>
normal line has negative reciprocal slope of tangent line therefore
![\displaystyle m_{ \rm normal} = \frac{5}{8}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%20%20m_%7B%20%5Crm%20normal%7D%20%3D%20%20%20%20%20%5Cfrac%7B5%7D%7B8%7D)
once again we'll use point-slope form of linear equation to get the equation and to get so substitute what we got
![\rm \displaystyle y - ( - 5a)= \frac{5}{8} (x - \frac{25}{16} a)](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20-%20%28%20%20-%205a%29%3D%20%20%20%5Cfrac%7B5%7D%7B8%7D%20%28x%20-%20%20%5Cfrac%7B25%7D%7B16%7D%20a%29)
simplify which yields:
![\rm \displaystyle y = \frac{5}{8} x - \frac{765}{128} a](https://tex.z-dn.net/?f=%20%5Crm%20%5Cdisplaystyle%20y%20%3D%20%20%5Cfrac%7B5%7D%7B8%7D%20x%20%20-%20%20%5Cfrac%7B765%7D%7B128%7D%20a)
and we're done!
( please note that "a" can't be specified and for any value of "a" the equations fulfill the conditions)
Answer:
point r (2,-9)
Step-by-step explanation:
You move left one and down 4 from point c