For a point P on an ellipse, let d be the distance from the center of the ellipse to the line tangent to the ellipse at P . Prove that ( P F 1 ) ( P F 2 ) d 2 is constant as P varies on the ellipse, where P F 1 and P F 2 are the distances from P to the foci F 1 and F 2 of the ellipse.
For a point P on an ellipse, let d be the distance from the center of the ellipse to the line tangent to the ellipse at P . Prove that ( P F 1 ) ( P F 2 ) d 2 is constant as P varies on the ellipse, where P F 1 and P F 2 are the distances from P to the foci F 1 and F 2 of the ellipse.
Solution Summary: The author explains how the expression (PF_1)d2 is constant as P varies on the ellipse.
For a point P on an ellipse, let d be the distance from the center of the ellipse to the line tangent to
the ellipse at P. Prove that
(
P
F
1
)
(
P
F
2
)
d
2
is constant as P varies on the ellipse, where
P
F
1
and
P
F
2
are the distances from P to the foci
F
1
and
F
2
of the ellipse.
2
Graph of h
6. The graph of the function h is given in the xy-plane. Which of the following statements is correct?
, the graph of h is increasing at an increasing rate.
(A) For
(B) For
(C) For
苏|4 K|4
π
π
, the graph of h is increasing at a decreasing rate.
2
0 and b>1
(B) a>0 and 01
(D) a<0 and 0
3.
Consider the sequences of functions fn: [-T, π] → R,
sin(n²x)
n(2)
n
(i) Find a function f : [-T, π] R such that fnf pointwise as
n∞. Further, show that f uniformly on [-T,π] as n→ ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7,π]?
Justify your answer.
[10 Marks]
Good Day,
Please assist with the following.
Regards,
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