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COORDINATE GEOMETRY- PARABOLAS WHOSE AXES ARE PARALLEL TO EITHER X-AXIS OR Y-AXIS

Posted August 23rd, 2010 by alpha
in

Q) The equation of the parabola is given : $ y^2-4y+4x=0 $.
Find : vertex , focus , axis , directrix.
In general , how do u find all these (as written above) in case of those parabolas whose axes are parallel either to X-axis or Y-axis?
Please show each and every step.( Do not skip any part because I'm a beginner in this section)
Urgent !!! Please !!!!!!!!

Thanks in advance!

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Comments

has vertex point O(0,0)

On August 25th, 2010 Structure says:

$ y^2=4px $ has vertex point O(0,0) axis x=0, y=0, focus F(p,0) and directrix x=-p
you have $ y^2-4y=-4x $ or $ y^2-4y+4=-4x+4 $ or $ (y-2)^2=-4(x-1) $ so vertex is V(1,2)
axis are x=1 and y=2 focus is F(0,2) and directrix x=2
You have

$$y^2-4y+4=-4x+4$$
$$x^2+y^2-4y+4=x^2-4x+4$$
$$x^2+(y-2)^2=(x-2)^2$$

The last relation shows that distance from a point P(x,y) to line x=2 is the same as the distance to the focus (0,2)

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