George Saliaris
Junior Member
- Joined
- Dec 15, 2019
- Messages
- 53
Can you please "type" your problem in this case. I have difficulty in deciphering your handwriting.I have considered using C-S, triangular inequality but nothing.. I am asked to proove the inequality using 'point distance from a line.'Any ideas?
Hint:abs(3a+4b+2)=5, prove that a^2+b^2+4b+7>=4a.
I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.You might start by completing the square in the inequality; it represents the exterior of a circle.
Then consider any point on either of the two parallel lines represented by the equation, and find its distance from the center of the circle. It should always be at least equal to the radius.
On the other hand, have you tried doing exactly what it says? Show that the distance from the center of the circle to the line is at least the circle's radius. If you can't do that, at least show us how you have learned to find such a distance.
Yeh ik that. P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.Hint:
a^2+b^2+4b+7>=4a
a^2+b^2+4b+7- 4a >= 0 ............ complete square
(a - 2)2 + (b + 2)2 >= 1
What does the following mean geometrically:I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.
Yeh ik that. P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.
P.S : This probem is from a Romanian Mathematics competition.
*Maybe the inequality has to do with 'other chapters' *
Do you not see the relationship between knowing the center of the circle and the rest of the problem? Any point farther from the center of the circle than its radius will satisfy the inequality. You can use the formula for the distance from a point to a line to show that every point on the line is farther than the radius. So if you do what I suggested, you will be doing what you were told to do, as I understand it. I think you will like the result. But in order to do that, you have to start! What work have you done?I am supposed to prove this inequality using the formula for 'point distance from a line, not a circle or some other kind of stuff.
P. S: I had proved this inequality like 1 year ago using asomewhat a lot of calculations.. But this year, my teacher said to me that using the 'formula for point distance from a line' the inequality has a truly 'unique' solution.