Question

If the curves ax2 +by2 = 1 and cx2 +dy2 =1 intersect each other orthogonally then show that
1/a -1/b = 1/c -1/d​

Answer 1

Step-by-step explanation:

Let (x

1

,y

1

) be the point of intersection of the curves. So, it satisfies both the curves,

ax

1

2

+by

1

2

=1 (1)

And,

cx

1

2

+dy

1

2

=1 (2)

From equation (1) and equation (2),

x

1

2

=

ad−bc

d−b

and y

1

2

=−

ad−bc

c−a

.

Differentiating equation (1) with respect to x

1

,

dx

1

dy

1

=−

by

1

ax

1

Differentiating equation (2) with respect to x

1

,

dx

1

dy

1

=−

dy

1

cx

1

Since, the slopes are cut each other orthogonally, then

−

by

1

ax

1

×−

dy

1

cx

1

=−1

bdy

1

2

acx

1

2

=−1

bd

ac

×−

c−a

d−b

=−1

bd

d−b

=

ac

c−a

b

1

−

d

1

=

a

1

−

c

1

a

1

−

b

1

=

c

1

−

d

1

Hence proved.