Straight lines, circles and now parabola, It always forms a sequence

Geometry Level pending

A circle touches the parabola y 2 = 4 a x y^{2}=4ax , at P. It also passes through focus S of parabola and intersects its axis at Q. The angle SPQ is 90 degrees .Then the equation of circle is A ( x ) 2 + B ( y ) 2 + C a ( x ) + D a ( y ) + E ( a ) 2 = 0 A( x )^{2} + B(y)^{2} + Ca(x)+Da(y)+ E(a)^2=0 .

Find the value of A + B + C + D + E A+B+C+D+E + ( Sum of numbers from 1 to 100 that are divisible by 2 or 5 )


The answer is 3051.

Aakash Khandelwal by Aakash Khandelwal , 21, India

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1 solution

Aakash Khandelwal
Aug 29, 2015

Take the point Q as (c,0) and write the diametric form of circle with point Q and focus of parabola as angle SPQ is right angled. Now put y^2 as 4ax and form a quadratic in x . Now circle should touch points at same absicca discriminant of this quadratic should be zero. There fore c= 3a . The rest part is self explanatory.

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