# How to find the coordinates?

Feb 25, 2018

$\left(x , y\right) = \left(\frac{4}{9} , 2\right)$

#### Explanation:

$4 y = 9 x + 4$
${y}^{2} = 9 x$
All we need to do is to solve the system of equations, the solutions are the required points, so:
$9 x = 4 y - 4$
$x = \frac{4 y - 4}{9}$
${y}^{2} = 9 \cdot \frac{4 y - 4}{9}$
${y}^{2} - 4 y + 4 = 0$
${\left(y - 2\right)}^{2} = 0$
$y = 2$
$x = \frac{4}{9}$
thus the required coordinated are:
$\left(x , y\right) = \left(\frac{4}{9} , 2\right)$

Feb 25, 2018

$\left(x , y\right) \to \left(\frac{4}{9} , 2\right)$

#### Explanation:

$4 y = 9 x + 4 \to \left(1\right)$

${y}^{2} = 9 x \to \left(2\right)$

$\text{from equation } \left(1\right) \textcolor{w h i t e}{x} 9 x = 4 y - 4 \to \left(3\right)$

$\text{substitute "9x=4y-4" in equation } \left(2\right)$

${y}^{2} = 4 y - 4 \leftarrow \textcolor{b l u e}{\text{rearrange and equate to zero}}$

$\Rightarrow {y}^{2} - 4 y + 4 = 0$

$\Rightarrow {\left(y - 2\right)}^{2} = 0 \Rightarrow y = 2$

$\text{substitute "y=2" in equation } \left(3\right)$

$9 x = 8 - 4 \Rightarrow x = \frac{4}{9}$

$\text{point of contact } = \left(\frac{4}{9} , 2\right)$
graph{(y-9/4x-1)(y^2-9x)=0 [-10, 10, -5, 5]}