Question

What is the curvature of x2 + y2 +4x−6y−1=0

Answer 1

Step-by-step explanation:

Given curve is x

2

+y

2

−4x−6y+9=0

⟹(x−2)

2

+(y−3)

2

−4−9+9=0

⟹(x−2)

2

+(y−3)

2

=4 which is a circle with centre (2,3) and radius 2

Since, one of the diameter of the above circle is chord of the other circle. So, length of the chord is 4

Draw a line joining the points (1,1) and (2,3)

Now, distance between the point (1,1) and (2,3) is given by

D=

(1−2)

2

+(1−3)

2

=

1+4

=

5

Now, draw a line segment joining the centre of new circle and one of the end of the chord of same circle.

So, the line obtained forms radius of the new circle

The point drawn from the centre to the mid-point of chord forms a right angle triangle.

So, we use Pythagoras theorem to get the radius of next circle

∴r

2

=(

5

)

2

+2

2

=5+4=9

∴r=3

Hence, radius of the new circle is 3.