ParabolamediumPYQ · JEE Main · 6 Apr 2026 · Shift 1 (Morning)Free

Focal-Angle Chord: sin α = 3/5 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question
Let chord PQPQ of length 3133\sqrt{13} of the parabola y2=12xy^2=12x be such that the ordinates of points PP and QQ are in the ratio 1:21:2. If the chord PQPQ subtends an angle α\alpha at the focus of the parabola, then sin⁡α\sin\alpha is equal to
A35\dfrac35correct
B45\dfrac45
C513\dfrac{5}{13}
D1213\dfrac{12}{13}
Solution
Step 1: y2=12x=4(3)xy^2=12x=4(3)x ∴ point (3t2,6t)(3t^2,6t). Take P(3t2,6t)P(3t^2,6t), Q(3t22,6t2)Q(3t_2^2,6t_2). Ordinate ratio 6t:6t2=1:2⇒t2=2t⇒Q(12t2,12t)6t:6t_2=1:2\Rightarrow t_2=2t\Rightarrow Q(12t^2,12t). Step 2:
PQ2=(12t2−3t2)2+(12t−6t)2=(9t2)2+(6t)2=81t4+36t2.PQ^2=(12t^2-3t^2)^2+(12t-6t)^2=(9t^2)^2+(6t)^2=81t^4+36t^2.
 PQ=313⇒PQ2=117PQ=3\sqrt{13}\Rightarrow PQ^2=117:
81t4+36t2=117⇒9t4+4t2−13=0⇒(t2−1)(9t2+13)=0.81t^4+36t^2=117\Rightarrow9t^4+4t^2-13=0\Rightarrow(t^2-1)(9t^2+13)=0.
 9t2+13>0⇒t2=1⇒t=19t^2+13>0\Rightarrow t^2=1\Rightarrow t=1. ∴ P(3,6), Q(12,12)P(3,6),\ Q(12,12). Step 3: Focus S(3,0)S(3,0). ∙\bullet SPSP: (3,0)(3,0) to (3,6)(3,6) ∴ vertical. ∙\bullet SQSQ: (3,0)(3,0) to (12,12)(12,12), slope =12−012−3=43=\dfrac{12-0}{12-3}=\dfrac43. SPSP vertical ⇒tan⁡α=912=34\Rightarrow\tan\alpha=\dfrac{9}{12}=\dfrac34. Step 4: tan⁡α=34⇒sin⁡α=35\tan\alpha=\dfrac34\Rightarrow\sin\alpha=\dfrac35 Correct answer: (1)
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