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Range of Ordinate of C with Right Angle on y²=x+4 | JEE

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

Question
Let A(0,2)A(0, 2), BB and CC be points on the parabola y2=x+4y^2 = x + 4 such that ∠CBA=π2\angle CBA = \dfrac{\pi}{2}. Then the range of the ordinate of CC is:
A(−∞,0)∪(4,∞)(-\infty, 0) \cup (4, \infty)
B(−∞,0]∪[4,∞)(-\infty, 0] \cup [4, \infty)correct
C[0,4][0, 4]
D(−∞,0)∪[4,∞)(-\infty, 0) \cup [4, \infty)
Solution
Step 1: Take B(t12−4,t1)B(t_1^2 - 4, t_1) and C(t2−4,t)C(t^2 - 4, t) on y2=x+4y^2 = x + 4, with A(0,2)A(0, 2). Then
slope BA=12+t1,slope BC=1t1+t.\text{slope }BA = \frac{1}{2 + t_1},\qquad \text{slope }BC = \frac{1}{t_1 + t}.
 Step 2: ∠CBA=90∘⇒\angle CBA = 90^\circ \Rightarrow product of slopes =−1= -1:
t12+(2+t)t1+(2t+1)=0.t_1^2 + (2 + t)t_1 + (2t + 1) = 0.
 Step 3: Real t1t_1 needs a non-negative discriminant.
t2−4t≥0⇒t∈(−∞,0]∪[4,∞).t^2 - 4t \ge 0 \Rightarrow t \in (-\infty, 0] \cup [4, \infty).
 Correct answer: (2)
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