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Let A,B, and C be three sets such that A={(x,y)|(x)/(cos theta)=(y)/(sintheta)=5,"where" 'theta'"is parameter"} B= {(x,y)|(x-3)/(cos phi)=(y-4)/(sin phi)=r} C= { (x,y)|(x-3)^(2)+(y-4)^(2)leR^(2)} If A capC =A, then minimum value of R is

Answer»

5
6
10
11

Solution :`A= { (x,y)}(x)/(cos theta) =(y)/(sin theta) =5, ` where `theta ` is parameter `}`
A is SET of points which lie of the CIRCLE `C_(1) : x^(2)+y^(2) =25`
`B = { (x,y)|(x-3)/(cos phi)=(y-4)/(sin phi) =r}`
If` phi` varies and r is fixed, then B is the set of all points which lie on a circle `C_(2) : (x-3)^(2) +(y-4)^(2) =r^(2)`
And if `phi` is fixed and r varies, then B is the set of all points which lie on the straight line `L : y-4= tan theta ( x-3)`.

Now, if `A cap C =A`
Both circle `C_(1)` and `C_(2)` must touch each other internally.
So, R must be 10.
If `phi` is fixed and r varies and `n ( A cap B) = 1`, circle `C_(1)` touches line L.
`phi =90^(@) + alpha`
`:. SEC phi = sec ( 90^(@) +alpha)= - cosect alpha = - (5)/(4)`


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