+0

# Find the most left and right tangent ?

0
360
2
+97

The question:

y=3.48x^2, determine the positionson the curve (x,y) of the left most tangent and the right most tangent that pass through the point (-5.67, -44.75).

If there is no tangent for either the left or right, just say none.

Round to 4 dec. places.

Thanks

math
Oli96  Aug 22, 2014

#3
+18827
+5

Find the most left and right tangent

$$\boxed{y=a*x^2\quad a=3.48}\\ \mbox{Point }(x_p,y_p) \text{ : } \left( x_p = -5.67 \mbox{ , }y_p =-44.75 \right)\\ \text{Find the tanget Point } (x_t, y_t) \\ \\ y_t = a*x_t^2\\ \text{Slop of } y=a*x^2 \text{ is } y'= 2a*x_t \\ \text{Slop of the line through Point p is } y' = \dfrac{y_t-y_p}{x_t-x_p}\\ \text{the slops must be equal: } y' = 2a*x_t = \dfrac{y_t-y_p}{x_t-x_p}\\\\ \Rightarrow 2a*x_t*(x_t-x_p)=y_t-y_p \quad | \quad y_t=a*x_t^2 \\ 2a*x_t*(x_t-x_p)=a*x_t^2-y_p \\ 2a*x_t^2 - 2a*x_t*x_p=a*x_t^2-y_p \\ 2a*x_t^2 -a*x_t^2 - 2a*x_t*x_p + y_p = 0 \\ \boxed{a*x_t^2 - 2a*x_t*x_p + y_p = 0 }\\$$

$$\Rightarrow \boxed{ x_{t_{1,2}}=x_p\pm\sqrt{x_p^2-\dfrac{y_p}{a}} \qquad y_{t_{1,2}}=a*x_{t_{1,2}}^2}$$

$$x_{t_1}=-5.67+\sqrt{(-5.67)^2-\left(\dfrac{-44.75}{3.48}\right) }\\ x_{t_1}=-5.67+6.70880730103 = 1.03880730103 \\ y_{t_1}=3.48*1.03880730103 ^2 = 3.75533971815$$
right most tangent

$$x_{t_2}=-5.67-\sqrt{(-5.67)^2-\left(\dfrac{-44.75}{3.48}\right) }\\ x_{t_2}=-5.67-6.70880730103 = -12.3788073010\\ y_{t_2}=3.48*(-12.3788073010)^2 = 533.257348282$$
left most tangent

heureka  Aug 22, 2014
Sort:

#2
+80866
+5

This one is a little tough!!

The slope of a tangent line to the given parabola at any point is just y' = 6.96x

Now, what we're looking for is at least one point on the parabola where the line through  (-5.67, - 44.75) is tangent to that point (or points).

Let's call the point(s) on the parabola (x, 3.48x^2). And the slope of the tangent line at that point is just 6.96x.

So, using this point on the parabola and the point (-5.67 , - 44.75), we have that, using the slope "formula,"

(3.48x^2 + 44.75) / (x + 5.67) = 6.96x

(3.48x^2 + 44.75)  /(x + 5.67) - 6.96x = 0

And solving this equation using the onsite calculator, we have....

$${\frac{\left({\mathtt{3.48}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{44.75}}\right)}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5.67}}\right)}}{\mathtt{\,-\,}}{\mathtt{6.96}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3\,406\,662\,741}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{49\,329}}\right)}{{\mathtt{8\,700}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3\,406\,662\,741}}}}{\mathtt{\,-\,}}{\mathtt{49\,329}}\right)}{{\mathtt{8\,700}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{12.378\: \!807\: \!301\: \!025\: \!932}}\\ {\mathtt{x}} = {\mathtt{1.038\: \!807\: \!301\: \!025\: \!932}}\\ \end{array} \right\}$$

And the equation of the line that goes through (-5.67, -44.75) and touches the parabola at 1.038807301025932 is given by

y +44.75 = 6.96(1.038807301025932)(x + 5.67)

y = 7.23009881514048672x + 40.9946602818465597024 - 44.75

y = 7.23009881514048672x -3.7553397181534402976 ........ and rounding, we have

y = 7.23x - 3.755

And the equation of the line that touches the graph at - 12.378807301025932 is given by

y + 44.75 = 6.96( - 12.378807301025932)(x + 5.67)

y = -86.15649881514048672x -488.5073482818465597024 - 44.75

y = -86.15649881514048672x - 533.2573482818465597024  .....and rounding, we have

y = -86.156x - 533.257

A graph of the solution is found here.........https://www.desmos.com/calculator/zoxdwcc43a

Whew!!!   That one was pretty challenging!!!

CPhill  Aug 22, 2014
#3
+18827
+5

Find the most left and right tangent

$$\boxed{y=a*x^2\quad a=3.48}\\ \mbox{Point }(x_p,y_p) \text{ : } \left( x_p = -5.67 \mbox{ , }y_p =-44.75 \right)\\ \text{Find the tanget Point } (x_t, y_t) \\ \\ y_t = a*x_t^2\\ \text{Slop of } y=a*x^2 \text{ is } y'= 2a*x_t \\ \text{Slop of the line through Point p is } y' = \dfrac{y_t-y_p}{x_t-x_p}\\ \text{the slops must be equal: } y' = 2a*x_t = \dfrac{y_t-y_p}{x_t-x_p}\\\\ \Rightarrow 2a*x_t*(x_t-x_p)=y_t-y_p \quad | \quad y_t=a*x_t^2 \\ 2a*x_t*(x_t-x_p)=a*x_t^2-y_p \\ 2a*x_t^2 - 2a*x_t*x_p=a*x_t^2-y_p \\ 2a*x_t^2 -a*x_t^2 - 2a*x_t*x_p + y_p = 0 \\ \boxed{a*x_t^2 - 2a*x_t*x_p + y_p = 0 }\\$$

$$\Rightarrow \boxed{ x_{t_{1,2}}=x_p\pm\sqrt{x_p^2-\dfrac{y_p}{a}} \qquad y_{t_{1,2}}=a*x_{t_{1,2}}^2}$$

$$x_{t_1}=-5.67+\sqrt{(-5.67)^2-\left(\dfrac{-44.75}{3.48}\right) }\\ x_{t_1}=-5.67+6.70880730103 = 1.03880730103 \\ y_{t_1}=3.48*1.03880730103 ^2 = 3.75533971815$$
right most tangent

$$x_{t_2}=-5.67-\sqrt{(-5.67)^2-\left(\dfrac{-44.75}{3.48}\right) }\\ x_{t_2}=-5.67-6.70880730103 = -12.3788073010\\ y_{t_2}=3.48*(-12.3788073010)^2 = 533.257348282$$
left most tangent

heureka  Aug 22, 2014

### 12 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details