?? equation tangent to curve... (1 Viewer)

[Mr Chi]

I need help with this question...

Find the equation of the tangent to the curve xsquared=2y at the point (4,-8). This tangent meets the directrix at point M. Find co-ordinates of M.

so far i have an equation 4x+y-8=0....where do i go from here, if that is correct.

 

[sasquatch]

For this question i got 4x - y - 24 = 0 as the equation.

Try again...(maybe i stuffed up..but i dont think i did..or hope i didnt).

After this you need to find the point where that curve intersections with the directrix of the parabola x² = 2y. A general equation for the parabola is given as, x² = 4ay, where a is the focal length. So from this you may find out the focal length of the parabola you are given. If you dont already know the equation of the directrix is y = -a, and therefore using simultaneous equations you may find the point of intersection (or point M) of the tangent and the directrix.

If your still stuck, tell me what on and ill try and help...

 

[XcarvengerX]

Solution.

Spoiler

Equation of the tangent can be found by make y the subject to the curve equation and use dy/dx. m = 4.
Use y - y₁ = m (x - x₁) to find the equation of the tangent. The equation is 4x - y - 24 = 0.
Next find the focus of the curve, and as the general formula for parabola is x² = 4ay, where (0,a) is the focus and y = -a is the directrix. In this case a = 2
Find the point of intersection using simultaneous equation, i.e. substitute this y-value of directrix to the equation of the tangent.
Solution: M(5.5,-2)