Question

`(x-7)² + (y+24)² = 196` Min`(x²+y²)=` ?

Answer 1

See below.

Explanation:

`(x_0,y_0) = (77/25, -264/25)`

This is the tangency point between the circle

`x^2+y^2=11^2`

and the circle

`(x-7)^2+(y+24)^2=14^2`

Answer 2

`121.`

Explanation:

Observe that, `S : (x-7)^2+(y+24)^2=196,` represents a circle

with Centre at `C( 7,-24)` and Radius `r=sqrt196=14.`

The Parametric Eqns. of `S` are given by,

`x=7+14costheta, y=-24+14sintheta, theta in [0,2pi).`

`:. x^2+y^2=(7+14costheta)^2+(-24+14sintheta)^2,`

`=(7^2+2*7*14costheta+14^2cos^2theta)`

`+(24^2-2*24*14sintheta+14^2sin^2theta),`

`=(7^2+24^2)+2*14(7costheta-24sintheta)+14^2(1),`

`=25^2+14^2+28*25(7/25costheta-24/25sintheta).`

Letting, `7/25=cosalpha," so that, "sinalpha=24/25,` we find,

`x^2+y^2=25^2+14^2+25*28cos(theta+alpha).`

Since, the Minimum Value of `cos" function is, "-1,`

We conclude that the Reqd. Minima is, `25^2+14^2-25*28`

`=25^2-2*25*14+14^2=(25-14)^2=121.`

Enjoy Maths.!