Sir, is it possible that, In the first question, As A & B are the mid points and the line joining the midpoints in an Equilateral triangle must pass from the center of the triangle. If we consider this, then we get MN as Diameter of the circle.
Vijay, the line joining the midpoints will not pass through the centre – you are possibly confusing with the median! If we join all three midpoints we will get another smaller similar (i.e. equilateral) triangle – see this post, last part https://crackthecat.wordpress.com/2013/08/06/the-basic-proportionality-theorem/
How can we use the Golden Ratio directly for the 2nd question.
Once we have proved it (which we have above), then in future we can directly use it because we know it will be applicable! Proving stuff to yourself is a key thing in Geometry – it helps you apply the right theorem/formula/concept at the right time.
CAT-holics 
SIR, there is a typo error in ques 1… it should be …. let AM be x, and not MN be x
Anand, I’ve corrected it now. Thanks for pointing out! 🙂
regards
J
I think the answer to 2nd part of 2nd question should be 1+sqrt5/2 not minus. Please take a look.
Regards,
K 😉
The ratio should be less than 1….(1+rt5)/2 gives the ratio in the reverse direction.
regards
J
ok 🙂 , so we have to pick the roots of the quadratic carefully .
Don’t we always? 🙂 Here’s another such case where we have to be cautious…one of my earliest posts. https://crackthecat.wordpress.com/2013/01/15/recursive-expressions-1/
regards
J
Sir, is it possible that, In the first question, As A & B are the mid points and the line joining the midpoints in an Equilateral triangle must pass from the center of the triangle. If we consider this, then we get MN as Diameter of the circle.
Vijay, the line joining the midpoints will not pass through the centre – you are possibly confusing with the median! If we join all three midpoints we will get another smaller similar (i.e. equilateral) triangle – see this post, last part https://crackthecat.wordpress.com/2013/08/06/the-basic-proportionality-theorem/
regards
J
Sir,
In the 2nd sum, with prior knowledge of golden ratio, we could have easily said that the answer would be the reciprocal of the golden ratio!
Exactly! Which is why prior knowledge is sometimes such a useful thing… 🙂
regards
J
Hi J,
How can we use the Golden Ratio directly for the 2nd question.
Once we have proved it (which we have above), then in future we can directly use it because we know it will be applicable! Proving stuff to yourself is a key thing in Geometry – it helps you apply the right theorem/formula/concept at the right time.
regards
J