Question

If $ x - y - 3 = 0 $ is a normal drawn through the point $ (5, 2) $ to the parabola $ y^2 = 4x $, then the slope of the other normal that can be drawn through the same point to the parabola is?

• A. 0
• B. -1
• C. 2
• D. -2

Hint:

For normals to parabola, use parametric forms and quadratic relations to find slopes.

Answer 1

The Correct Option is D

Equation of normal to \( y^2 = 4ax \) is: \[ y = m x + \frac{a}{m} \] Given one normal passes through \( (5, 2) \) with slope \( m_1 \). Use point-slope form and solve for other slope \( m_2 \). Given \( x - y - 3 = 0 \) implies slope \( m_1 = 1 \). Using normal condition and point substitution, solve quadratic in \( m \). Other slope \( m_2 = -2 \).