Explanation
1. Identify Parabola Parameters
The given parabola is y2=36x.
Comparing with y2=4ax:
2. Determine the Slope of PQ
The length of a focal chord L making an angle θ with the x-axis is:
Substitute L=100 and a=9:
100=36csc2θ⟹csc2θ=36100=925
sinθ=53⟹cosθ=54 (since θ is acute)
Slope of PQ (m1) =tanθ=43.
3. Locate Point M
For a focal chord PQ, let the focus be S(9,0). The distances SP and SQ satisfy:
SP+SQ=100 and SP⋅SQ=1L⋅a (or use harmonic mean property SP1+SQ1=a1=91).
Solving these: SP=90 and SQ=10.
Point M divides PQ in the ratio PM:MQ=3:1.
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PM=75 and MQ=25.
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Since SP=90, point M lies on SP at a distance of SP−PM=15 units from the focus S toward P.
Using the parametric form from focus S(9,0):
xM=9+15cosθ=9+15(54)=21
Thus, M=(21,9).
4. Equation of the Perpendicular Line
The slope of PQ is m1=43. The slope of the line perpendicular to PQ is m2=−34.
The equation of the line passing through M(21,9) with slope −34 is:
5. Verify Options
We check which point does NOT satisfy 4x+3y=111:
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(1) (3, 33): 4(3)+3(33)=12+99=111 (Lies on the line)
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(2) (6, 29): 4(6)+3(29)=24+87=111 (Lies on the line)
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(3) (-6, 45): 4(−6)+3(45)=−24+135=111 (Lies on the line)
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(4) (-3, 43): 4(−3)+3(43)=−12+129=117=111 (Does NOT lie on the line)
Correct Option: (4)
