Question

Show that the curves 4x = y2 and 4xy=k cut at right angles, if k2 =512.
​

Answer 1

Answer:

ANSWER

The given curves are,

⇒ 4x=y

2

----- ( 1 )

⇒ 4xy=k ----- ( 2 )

We have to prove that two curves cut at right angles if k

2

=512

Now,

4x=y

2

Differentiating both sides w.r.t.x, we get

⇒ 4=2y.

dx

dy

⇒

dx

dy

=

y

2

⇒ m

1

=

y

2

----- ( 3 )

4xy=k

Differentiating both sides w.r.t.x, we get

⇒ 4(1×y+x

dx

dy

)=0

⇒ y+x

dx

dy

=0

⇒

dx

dy

=

x

−y

⇒ m

2

=

x

−y

----- ( 4 )

It is given that two curves intersect orthogonally.

∴ m

1

.m

2

=−1

⇒

y

2

×

x

−y

=−1 [ From ( 3 ) and ( 4 ) ]

⇒

x

−2

=−1

⇒ x=2

Now,

4xy=k

⇒ (y

2

)y=k [ Since, 4x=y ]

⇒ y

3

=k

⇒ y=k

3

1

Substituting y=k

3

1

in equation ( 1 ) we get,

⇒ 4x=(k

3

1

)

2

⇒ 4×2=k

3

2

⇒ 8=k

3

2

⇒ k

2

=(8)

3

[ Taking cube on both sides ]

⇒ k

2

=512