A geometry problem by avi solanki
A parabola is passing through four points ( 0 , 1 ) , ( 0 , 2 ) , ( 2 , 2 ) , ( 2 , 0 ) If the slope of its axis of symmetry is m and ∣ m ∣ = b a , where a and b are coprime positive integers, find a + b .
The answer is 5.
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4 solutions
Wonderful.This is the shortest solution. The locus of the midpoints of a system of parallel chords (of slope m) drawn to a parabola (y^2=4ax) is the diameter of the parabola (y=2a/m) which is parallel to the axis of symmetry. Thanks for sharing the solution.
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what do u mean by "diameter" here ?
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I have written that in my comment.It is the locus of the midpoints of a system of parallel chords drawn to the parabola and has the equation y=2a/m in case of a standard parabola, where m is the slope of the system of parallel chords.
WOW! I am a duffer. I used a x 2 + b y 2 + 2 h x y + 2 g x + 2 f y + c = 0 and found out all the constants to do this question. :P
Let equation of the parabola be h²x²+2hxy+y²+mx+ny+c=0. On substituting x=0 we get quadratic equation in y that must have roots 1 and 2.So parabola becomes h²x²+2hxy+y²+mx-3y+2=0.Now substitute points (2,0) and (2,2) points in parabola to get 4h²+2m+2=0 (say eq 1) and 4h²+8h+4+2m-6+2=0, that is 4h²+8h+2m=0 (say eq 2).Subtracting equation 1 from 2 gives h=(1/4).Now, the angle through which the X axis must be rotated so that it becomes parallel to the axis of symmetry of the parabola satisfies the equation tan (2 theta)=2h/(a-b) [in usual notation of equation of general parabola].So we get that for the given parabola tan(2 theta)=2h/(h²-1)=(-8/15).So we get tan(theta)=(-1/4) or 4.Hence |m|=1/4 or 4 either of which gives the answer 5. And the equation of the parabola is x 2 + 8 x y + 1 6 y 2 − 1 8 x − 4 8 y + 3 2 = 0
thank u for the detailed solution @Indraneel Mukhopadhyaya
Consider a general line in slope intercept form to be the line about which the parabola is symmetric. Equate the perpendicular distances on the line from the pairs of points with abscissa 0 and those from abscissa 2 separately. Solve the equations for the slope and the y intercept. The line can be obtained as y = ( 1 / 4 ) x + ( 3 / 2 )
If you want a detailed proof with all the equations please let me know.
Also this is the shortest way I could think of.
Check out my proof. I have tried to give a detailed proof.
Maybe you have done some error in calculation, because the slope of axis of symmetry of parabola is 1/4 and not (-1/4).
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Oh yes sorry happened by mistake
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Sorry, I had made a calculation error.The axis of symmetry has slope (-1/4) and not (1/4).
I would suggest that you work out your proof again as your equation is wrong.I say this because the focus which is (-59/102 , 689/408) does not lie on your axis of symmetry.Also ,I am not sure about the correctness of your proof (why should the points with same abscissa be equidistant from the axis?). Note that the correct equation of axis is y=(-1/4)x + (105/68)
Since (0,1)-(0,2) and (2,2)-(2,0) form two parallel chords, the axis of symmetry is parallel to the midpoints of the chords..
So the slope through (0,3/2) and (2,1), same as that of the axis. That is m=(1 - 3/2)/(2 - 0) =- 1/4.
|m|=|- 1/4|=a/b. Thus a+b=1+4=5.
No sir. The axis of symmetry is parallel (and not coincident) to the line joining the midpoints of the parallel chords.
Let the four points have names: A ( 0 , 1 ) , B ( 0 , 2 ) , C ( 2 , 2 ) , D ( 2 , 0 ) The slope is the same as a slope between midpoints of two parallel segments A B : ( 0 , 3 / 2 ) and C D : ( 2 , 1 ) . The slope is: m = 2 − 0 1 − 3 / 2 = − 4 1