Question

find the point on the curve x2 +y2 -4xy +2 =0 where the normal is parallel to x-axis ​

Answer 1

Answer:

The equation of the circle is,

x

2

+y

2

−2x−4y+1=0...(i).

At points where the tangent is parallel to x-axis the slope is 0.

Differentiate the above equation with respect to x.

Thus,

2x+2yy

′

−2−4y

′

=0

⇒y

′

(y−2)=1−x

⇒y

′

=

y−2

1−x

Equate the above slope to 0.

The slope y

′

=

y−2

1−x

=0⇒x=1

Put the above value of x in (i) to get the corresponding values of y.

Thus,

1

2

+y

2

−2⋅1−4y+1=0

⇒y

2

−4y=0

y=0,4

Thus the 2 points are,

P(1,0)

Q(1,4)