1. ## equations intersections

hi, my question is: find the equations of the tangents to the curve with equation $\displaystyle y = (x + 1)(5 - x)$ at the points P(-1, 0) and Q(5, 0). prove that these two tangents intersect at the point (2, 18)

the first thing i did was expand the brackets and differentiate with x = -1 (for P). which resulted in:

$\displaystyle -x^2 + 4x + 5$ so then $\displaystyle \frac{dy}{dx} = - 2x + 4$ and usig x = -1 the gradient comes to 6

so the equation is $\displaystyle y - 0 = 6(x + 1) \implies y = 6x + 6$

so for Q i did the same thing. used x = 5 for $\displaystyle -2x + 4$ which gave a gradient of - 6

so the equation is $\displaystyle y - 0 = -6(x - 5) \implies y = -6x + 30$

how would i prove these two tangents intersect at point (2, 18)?

thanks, mark

2. Originally Posted by mark
hi, my question is: find the equations of the tangents to the curve with equation $\displaystyle y = (x + 1)(5 - x)$ at the points P(-1, 0) and Q(5, 0). prove that these two tangents intersect at the point (2, 18)

the first thing i did was expand the brackets and differentiate with x = -1 (for P). which resulted in:

$\displaystyle -x^2 + 4x + 5$ so then $\displaystyle \frac{dy}{dx} = - 2x + 4$ and usig x = -1 the gradient comes to 6

so the equation is $\displaystyle y - 0 = 6(x + 1) \implies y = 6x + 6$

so for Q i did the same thing. used x = 5 for $\displaystyle -2x + 4$ which gave a gradient of - 6

so the equation is $\displaystyle y - 0 = -6(x - 5) \implies y = -6x + 30$

how would i prove these two tangents intersect at point (2, 18)?

thanks, mark
Solve simultaneously:

y = 6x + 6 .... (1)

y = -6x + 30 .... (2)

And in case you're wondering, I Thanked your post because you showed all your working and you clearly stated where you got stuck (a rarity for this kind of question).

3. ah thanks. i really should have thought of solving them simultaneously. thanks for the help

4. actually hold on, i tried simultaneously solving them and it didn't turn out how i expected.

so its $\displaystyle 6x + 6 = -6x + 30$ which leaves $\displaystyle y = 12x - 24$

or i just added equation 1 and 2 together to cancel the x's and it left y = 36

could someone show me what to do from here please? my maths has gotten really rusty from not doing it for ages

thanks

5. ah don't worry i got it now