Question

PQ is a focal chord of the parabola \( y^2 = 4x \) with focus S. If \( P = (4,4) \), then SQ = ?

• A. 2
• B. \(\dfrac{5}{4}\)
• C. 5
• D. \(\dfrac{3}{2}\)

Hint:

In a parabola \( y^2 = 4ax \), a focal chord connects two points whose parameters are reciprocal. Use this property along with parametric coordinates to simplify chord-based problems.

Answer 1

The Correct Option is B

Step 1: Identify the properties of the parabola 
The standard equation of the parabola is \( y^2 = 4ax \). Comparing, we get \( a = 1 \). 
So, the focus \( S \) is at \( (a, 0) = (1, 0) \). 
Step 2: Use property of focal chord 
Let the point \( P = (4,4) \) be on the parabola. Its parametric form is \( (at^2, 2at) \). 
Since \( a = 1 \), comparing: 
\[ at^2 = 4 \Rightarrow t^2 = 4 \Rightarrow t = 2, \quad \text{and} \quad 2at = 4 \Rightarrow t = 2 \] For a focal chord, if one point is at \( t \), the other point \( Q \) will be at \( \frac{1}{t} \). So the parametric point \( Q = (a/t^2, 2a/t) = (1/4, 1) \) 
Step 3: Use distance formula to find SQ 
Focus \( S = (1, 0) \), \( Q = \left(\frac{1}{4}, 1\right) \) \[ SQ = \sqrt{\left(1 - \frac{1}{4}\right)^2 + (0 - 1)^2} = \sqrt{\left(\frac{3}{4}\right)^2 + 1} = \sqrt{\frac{9}{16} + \frac{16}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \] \[ \boxed{\frac{5}{4}} \]