# Thread: Line and hyperbola intersection

1. ## Line and hyperbola intersection

"Find the coordinates of the points of intersection of the hyperbola defined by ((x-1)^2)/4 - ((y+2)^2)/9 = 1 and the line 5x + 12y = 0."

This is really an easy question but for some reason I keep getting it wrong so I must be doing a small mistake somewhere and after several repetitions, I still cannot spot my mistake!

The answer is: (-1.69, 0.71) and (3.06, -1.27)

If anyone could please do it for me, I'd greatly appreciated it!

2. Originally Posted by s3a
"Find the coordinates of the points of intersection of the hyperbola defined by (1) ((x-1)^2)/4 - ((y+2)^2)/9 = 1 and the line (2) 5x + 12y = 0."

This is really an easy question but for some reason I keep getting it wrong so I must be doing a small mistake somewhere and after several repetitions, I still cannot spot my mistake!

The answer is: (-1.69, 0.71) and (3.06, -1.27)

If anyone could please do it for me, I'd greatly appreciated it!
1. From (2): $y = -\dfrac5{12}x$

2. Plug in this term for y into (1):

$\dfrac{(x-1)^2}4-\dfrac{\left( -\dfrac5{12}x + 2 \right)^2}9=1$

3. Expand the brackets:

$\dfrac{x^2-2x+1}4-\dfrac{ \dfrac{25}{144} x^2- \dfrac{20}{12} x + 4}9=1$

4. Multiply both sides by $9 \cdot 144 = 1296$

$324x^2-648x + 324 - 25x^2 +240x-576=1296$

and collect like terms:

$299 x^2 - 408x - 252 = 1296~\implies~299x^2 - 408x - 1548 = 0$

5. Now apply the quadratic formula:

$x = \dfrac{204}{299}\pm\dfrac{54\cdot \sqrt{173}}{299}$

with rounded values you get: $x = 3.057722768 ~\vee~ x = -1.693174273$

6. Plug in these values into (2) to calculate the y-values.