11
You visited us
11
times! Enjoying our articles?
Unlock Full Access!
Byju's Answer
Standard XII
Mathematics
Parametric Equation of Normal
Prove that th...
Question
Prove that the product of perpendiculars from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
to its asymptotes is constant and the value is
a
2
b
2
a
2
+
b
2
Open in App
Solution
The equation of asymptotes are
b
x
+
a
y
=
0
and
b
x
−
a
y
=
0
Let the point on hyperbola be
(
p
,
q
)
Perpendicular distance from the point
(
p
,
q
)
to the line
b
x
+
a
y
=
0
is
b
p
+
a
y
√
a
2
+
b
2
Perpendicular distance from point
(
p
,
q
)
to the line
b
x
−
a
y
=
0
is
b
p
−
a
q
√
a
2
+
b
2
The product of distances will be
b
2
p
2
−
a
2
q
2
a
2
+
b
2
The point
(
p
,
q
)
lie on hyperbola , so by substituting we get
b
2
p
2
−
a
2
q
2
=
a
2
b
2
Therefore the product of distances is
a
2
b
2
a
2
+
b
2
Hence proved
Suggest Corrections
0
Similar questions
Q.
The product of the perpendiculars from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
to its asymptotes is equal to
k
a
2
b
2
a
2
+
b
2
then
k
=
Q.
The product of the perpendiculars from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
to its asymptotes is
Q.
The product of the perpendicular from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
to its asymptotes, is equal to
Q.
If
P
1
and
P
2
are the perpendiculars from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
on its asymptotes, then :
Q.
If
P
1
and
P
2
are the perpendiculars from any point on the hyperbola
x
2
a
2
−
y
2
b
2
=
1
on its asymptotes, then :
View More
Join BYJU'S Learning Program
Grade/Exam
1st Grade
2nd Grade
3rd Grade
4th Grade
5th Grade
6th grade
7th grade
8th Grade
9th Grade
10th Grade
11th Grade
12th Grade
Submit
Related Videos
Line and a Parabola
MATHEMATICS
Watch in App
Explore more
Parametric Equation of Normal
Standard XII Mathematics
Join BYJU'S Learning Program
Grade/Exam
1st Grade
2nd Grade
3rd Grade
4th Grade
5th Grade
6th grade
7th grade
8th Grade
9th Grade
10th Grade
11th Grade
12th Grade
Submit
AI Tutor
Textbooks
Question Papers
Install app