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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)

B

and

C

be points on the parabola

y^2 = x + 4

such that

\angle CBA = \dfrac{\pi}{2}

. Then the range of the ordinate of

C

is:

(-\infty, 0) \cup (4, \infty)

(-\infty, 0] \cup [4, \infty)

correct

[0, 4]

(-\infty, 0) \cup [4, \infty)

Solution

Step 1: Take

B(t_1^2 - 4, t_1)

and

C(t^2 - 4, t)

y^2 = x + 4

, with

A(0, 2)

. Then

\text{slope }BA = \frac{1}{2 + t_1},\qquad \text{slope }BC = \frac{1}{t_1 + t}.

Step 2:

\angle CBA = 90^\circ \Rightarrow

product of slopes

= -1

t_1^2 + (2 + t)t_1 + (2t + 1) = 0.

Step 3: Real

t_1

needs a non-negative discriminant.

t^2 - 4t \ge 0 \Rightarrow t \in (-\infty, 0] \cup [4, \infty).

Correct answer: (2)

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