31. Prove that the line px + y + r = 0 touches the ellipse
?
=
if a_p2 + b²q2 = r.
b2
+
a?
Answers
Answered by
10
Step-by-step explanation:
Let the line px + qy + r = 0 ..(1)
touch the circle x
2
+ y
2
= a
2
.(2)
at the point (h, k).
Now the equation of tangent at (h, k) is
xh + yk - a
2
= 0 .(3)
Comparing (1) and (3), we get
p
h
=
q
k
=
r
−a
2
whence h = -a
2
p/r and k = a
2
q/r
Since (h , k) lies on the given circle ∴ from (2)
a
4
p
2
/r
2
+ a
4
q
2
/r
2
a
2
or r
2
= a
2
(p
2
+ q
2
)
which is the required condition and the point of
contact is (- a
2
p/r , - a
2
q/r).
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