Let AB be a fixed chord passing through the focus of a parabola. how many circles can be drawn which
touch the parabola and AB at the focus
62Dont think this is a very difficult question..
I will tell you what comes to my mind.. and then may be you can try using the same..
L1 =0 is the line through the origin.. it will be (x-a)+k(y) = 0
Then look at a circle of 0 radius at focus, C= (x-a)2+y2=0
the family of circles which have the point a,0 as tangent is given by C+λL1
Now you want this circle to be tangential to the parabola...
substitute y2=4ax
You get a quadratic in x...
What next?
Can someone think of a more geometric proof? or any other suggestions?
1HINT:the way i reacted to solve was first of all noting these things::
(a).i need to get a cubic equation with THREE real roots.
(b).only property i can use is that of COMMON TANGENTS
so now this question is killed.....
bhaiya i din give as it was tough but as to see how one shud get """FIRST THOUGHTS""" on seeing any problem
39The quadratic in x will have equal roots as it touches the parabola at only one point...so D = 0?
1nahin yaar pritish u din get wat NISHANT sir wrote u will get two
general equations of circle.....[infact they r identical......so equate coefficients]......+before this circle thing sir mentioned u should take SLOPE form of tangent to parabola!!!!!!