please help - tangent to curve parabola (1 Viewer)

[dearrachel]

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

I have found that the equation of the tagent is 4x + y - 8 = 0
Can someone help me with the second part of the question? Much appreciated

 

[tohriffic]

Note: You should always draw a diagram especially if you get stuck.

Okay so you know that <img src="http://latex.codecogs.com/gif.latex?a" title="a" /> is equal to 0.5. [ <img src="http://latex.codecogs.com/gif.latex?x^2 = -4ay" title="x^2 = -4ay" />]

Therefore the directrix is at <img src="http://latex.codecogs.com/gif.latex?y = \frac{1}{2}" title="y = \frac{1}{2}" /> (Because the directrix is on the other side of the parabola)

So, to find the point of intersection, just solve the equation of the tangent and the directrix.

I attached a graph of it in just in case you didn't really get what I was saying about the equation of the directrix.

http://imageshack.us/photo/my-images/254/parabolap.jpg/

 

[bleakarcher]

The given point does not lie on the parabola.

 

[SpiralFlex]

The equation of the parabola should be or the coordinate should be

However by analysing the equation of the tangent you have given us, I can conclude that the coordinates are wrong instead of the equation.

Now,

Always draw a pretty diagram,

It is a parabola with negative concavity. The equation is in the general form,

is the focal length.



[Focal length]

The vertex is obviously

So the directrix is above the vertex, holding an equation of

Now your tangent,



Substituting your y coordinate of your directrix,

 

[tohriffic]

Oh. That's awkward.

 

[bleakarcher]

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

I have found that the equation of the tagent is 4x + y - 8 = 0
Can someone help me with the second part of the question? Much appreciated

x^2=2y=4ay, where a=1/2
dy/dx=x
When x=4,
=>dy/dx=m(t)=4
Let the tangent have equation y-y1=m(x-x1).
:.y-8=4(x-4)=4x-16
ie y=4x-8
Now, the parabola has directrix y=-1/2.
When the tangent at the (4,8) meets the directrix:
=>4x-8=-1/2
x=15/8=1.875, y=-0.5
Hence, Q(1.875,-0.5).

 

[SpiralFlex]

x^2=2y=4ay, where a=1/2
dy/dx=x
When x=4,
=>dy/dx=m(t)=4
Let the tangent have equation y-y1=m(x-x1).
:.y-8=4(x-4)=4x-16
ie y=4x-8
Now, the parabola has directrix y=-1/2.
When the tangent at the (4,8) meets the directrix:
=>4x-8=-1/2
x=15/8=1.875, y=-0.5
Hence, Q(1.875,-0.5).

No minus sign for a half.