The set of all these points P is an ellipse.
P(x,y) is on the ellipse <=> P'(x, (a/b) y ) is on C <=> x2 + (a/b)2 y2 = a2 <=> x2 y2 ---- + ---- = 1 a2 b2The points A(a,0) A'(-a,0) B(0,b) B'(0,-b) are called the vertices of the ellipse.
If a = b then the ellipse is a circle.
x = a cos(t) y = b sin(t)are parametric equations of the ellipse.
One can construct, for every t, a point on the ellipse.
Draw a line parallel to the y-axis through the intersection point of OP' and the circle with radius a.
Draw a line parallel to the x-axis through the intersection point of OP' and the circle with radius b.
The intersection point of these lines is a point of the ellipse.
Say P(a cos(t) , b sin(t)) is a variable point of the ellipse.
|PF|2 = (a cos(t) - c)2 + b2 sin2(t) = a2 cos2(t) - 2 a c cos(t) + c2 + (a2 - c2) sin2(t) = a2 - 2 a c cos(t) + c2 cos2(t) = (a - c cos(t))2 |PF'|2 = ... = (a + c cos(t))2 Since a > c : |PF| + |PF'| = a - c cos(t) + a + c cos(t) = 2a = constantConversely, we show that point P is on the ellipse if |PF| + |PF'| = 2a.
Connect P with F and F'. Say P' in the intersection point of F'P with the ellipse. (see figure)
If P is not on the ellipse then
|PF| + |PF'| = 2a and |P'F| + |P'F'| = 2a
We have |PF| + |PF'| -|P'F| - |P'F'| = 0
Then, we have |P'P| + |PF| = |P'F| and this is impossible.
|If F and F' are the foci of een ellipse x2/a2 + y2/b2 = 1 , then we have for each point P of the ellipse |PF| + |PF'| = 2a.|
x2 y2 -- + -- = 1 a2 b2To obtain the slope of the tangent line we differentiate implicitly.
2x 2y y' -- + ----- = 0 a2 b2Solving for y', we obtain
b2 x y'= - ---- a2 ySay D(xo,yo) is a fixed point of the ellipse.
b2 xo y'= - ------ a2 yoThe equation of the tangent line is
b2 xo y - yo = - ----- (x - xo) a2 yo <=> a2 yo y - a2 yo2 = b2 xo2 - b2 xo x <=> a2 yo y + b2 xo x = a2 yo2 + b2 xo2 <=> since D(xo,yo) is on the ellipse a2 yo y + b2 xo x = a2 b2 <=> xo x yo y ---- + ---- = 1 a2 b2The last equation is the tangent line in point D(xo,yo) of an ellipse.
Since |D,F| = |D,F"| , |F',F"| = 2a .
Now in the triangle F'TF" , we see that
|F',T| + |T,F"| > 2a => |T,F'| + |T,F| > 2aAnd from the definition of ellipse, it follows that T is outside of the ellipse. Hence all the points of t, different from D, are outside of the ellipse and therefore the bisector t of the lines DF and DF' is a tangent line of the ellipse.
The given solution is not 'the' solution.
Most exercises can be solved in different ways.
It is strongly recommended that you at least try to solve the problem before you read the solution.
Find the equation of the normal line in a point P(xo,yo) of the ellipse
b2 x2 + a2 y2 = a2 b2
Take on the ellipse E a variable point P and F is F(c,0).|
Show that the following lines are concurrent.
P is a variable point on the ellipse E and F is F(c,0).|
d is the line x = a2/c (directrix associated to F).
Show that |PF|/|P,d| is constant.
P( a cos(t), b sin(t) ) is on the ellipse E.|
Calculate the distances |PF| en |PF'|
|Find the product of the distances from the foci of an ellipse to a fixed tangent line. Show that this product is constant.|
|In an ellipse E take all chords with slope m. Show that the centers of these chords are on one line.|
Line r has an equation 3x - 4y = 0|
Line r'has an equation 3x + 4y = 0
Find the locus of the points P such that
| P,r |2 + | P,r' |2 = 10
P is a variable point of the ellipse b2 x2 + a2 y2 =a2 b2.|
Let A = A(a,0) and A' = A'(-a,0).
M1 is on the x-axis such that APM1 is a right angled triangle in P.
M2 is on the x-axis such that A'PM2 is a right angled triangle in P.
Show that the distance |M1 M2| is constant.
|A variable tangent line to the standard ellipse forms with the x-axis and the y-axis a triangle. Find the tangent lines such that the area of the triangle is minimum.|