Question

A tangent is drawn to the parabola y² = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it ?

• A. (0, 3)
• B. (-6, 0)
• C. (5, 4)
• D. (4, 5)

Hint:

For parabola $y^2 = 4ax$, the tangent with slope $m$ is $y = mx + \frac{a}{m}$. For $x^2 = 4ay$, it is $y = mx - am^2$.

Answer 1

The Correct Option is C

Step 1: Given parabola $y^2 = 6x$, comparing with $y^2 = 4ax$, we get $4a = 6 \Rightarrow a = 3/2$.
Step 2: Slope of given line $2x + y = 1$ is $m_1 = -2$.
Step 3: Since the tangent is perpendicular, its slope $m = -1/m_1 = 1/2$.
Step 4: Equation of tangent: $y = mx + a/m \Rightarrow y = \frac{1}{2}x + \frac{3/2}{1/2} \Rightarrow y = \frac{1}{2}x + 3 \Rightarrow x - 2y + 6 = 0$.
Step 5: Checking points: $(5, 4) \Rightarrow 5 - 2(4) + 6 = 3 \neq 0$. Thus (5, 4) does not lie on it.