Is it really an inequality Problem?
If x 2 + y 2 + 1 0 x − 2 4 y − 2 7 = 0 then the minimum value of x 2 + y 2 .
The answer is 1.
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4 solutions
Sorry I didnt noticed that you also posted a solution.
Our solutions are same, I have just elaborated things in my solution.
Exactly Same Way. I loved it when I found a geometric proof for this. Is there any algebraic method? Cauchy-Schwarz is not helping
why you are writing in a book from bottom to top (instead of from top to bottom)
For those of you dying (or just plain ol' waiting) for an algebraic solution:
We can rewrite the equation as ( x + 5 ) 2 + ( y − 1 2 ) 2 = 1 4 2 or ( x + 5 ) 2 + ( y − 1 2 ) 2 = 1 4 . By Minkowski's Inequality ,
( x + ( 5 ) ) 2 + ( y + ( − 1 2 ) ) 2 ≤ x 2 + y 2 + 5 2 + ( − 1 2 ) 2
1 4 ≤ x 2 + y 2 + 1 3
x 2 + y 2 ≥ 1
So, our final answer is 1
I generalized my solution and came up with the exactly same inequality you used and was thinking that it was an invention, but it's Minkowski's Inequality. :(
We know that minimum distance of a point from the circle is equal to CP-r,where c is center,P origin(here) and r radius.
Completing the square, we can write the equation of the circle as ( x + 5 ) 2 + ( y − 1 2 ) 2 = 1 4 2 . The answer is 1 4 − 1 3 = 1 , the difference between the circle's radius and the distance of the circle's center from the origin.