Length of tangents - circles and ellipses

[Peter Pedant]

Proper maths q coming up

The reader will know that plugging in the co ordinate (p,q)
into the eqn of a circle when (p,q) does NOT lie on the circle
will return the length of a tangent to the circle


ellipses - I recollect dimly - if you do the same for an ellipse, that is plug in a point that doesnt lie on an ellipse into the equation what if anything does the number represent ?
....
and if you put the point (p,q) in to the standard eqn of a tangent
x x1/ a2 + y y`/b2 ......what is the significance of the line that you get

I think these were called polar equations but I cant find anything on the internet

Thanks

[jake-the-peg]

I think there is some confusion here - by definition a tangent just touches a circle at a single point

I think what you mean is that it returns the distance from the point to the closest point on the circles perimeter i.e the length of the Normal rather than the tangent?

So take the circle with a radius 3 x²+y²=9

if we take the point (0,1) then it's 2 units from the circle's perimeter at (0,3)

put this in to the left hand side and subtract the *Square Root* of the figure on the right and you get 1-3 = -2 which is what you're looking for

Not sure about an elipse - probably different as the above is so because the formula gives you the radius from a central point which obviously isn't true for an elipse

A bit of manipulation might get you a similar formula

Check out these resources:

http://www.mathopenref.com/ellipse.html (simple)
http://mathworld.wolfram.com/Ellipse.html (more complex)

[factor-fiction]

I love helping with Maths problems on here but I couldn't help on this occasion as I didn't understand the bit about "length of a tangent" so I'm not sure what you want.
This page may be a useful starting point
http://en.wikipedia.o...247099830034543036653

[Peter Pedant]

you're right I meant the length of a normal
thanks

[Peter Pedant]

no I didnt
to takethe example - x2 + y2 = 9 or x2+y2 -9 = 0
which is a circle of radius 3 centred on the origin

take a point (5,0) - so it is 5 along the x axis

plug it in and take the sqrt
so it is root (25-9) or rt16 or 4
and this is the length of the tangent from (5,0) to the circle
(also a 3,4,5 triangle)

and I was asking if anyone can remember the same result for ellipses ?

[vascop]

I'll look at this and report back.

[vascop]

It certainly works for an ellipse if the point is on the x axis. If the point is off the x-axis it gets more complicated because there are 2 tangents to the ellipse through the point and the distances from the point to where the tangent touches the ellipses is different.( ie 2 values)

[vascop]

Hi Peter
The equation in your post:
x x1/ a^2 + y y1/b^2 is the equation of the tangent to the ellipse at the point P(x1,y1) ON the ellipse. If, on the other hand, the point P(x1,y1) lies outside the ellipse then this equation is the equation of the line which passes through the 2 points where the 2 tangents from P(x1,y1) touch the ellipse. This line is called the polar of the point P with respect to the given ellipse and the point P is called the pole of the line.
The same is true for the circle.