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Mathematics
Mathematics
Problem
Show that the line
x
cos
α
+
y
sin
α
=
p
x \cos \alpha + y \sin \alpha = p
x
cos
α
+
y
sin
α
=
p
touches the ellipse
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
a
2
x
2
+
b
2
y
2
=
1
if
a
2
cos
2
α
+
b
2
sin
2
α
=
p
2
a^2 \cos^2 \alpha + b^2 \sin^2 \alpha = p^2
a
2
cos
2
α
+
b
2
sin
2
α
=
p
2
and that point of contact is
(
a
2
cos
α
p
,
b
2
sin
α
p
)
\left( \frac{a^2 \cos \alpha}{p}, \frac{b^2 \sin \alpha}{p} \right)
(
p
a
2
c
o
s
α
,
p
b
2
s
i
n
α
)
.
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