Question

Let O be the vertex of the parabola \(y^2 = 4x\) and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:

• A. 1
• B. 2
• C. 4
• D. 8

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are dealing with a standard parabola \( y^2 = 4ax \). We take two points P and Q on it such that they subtend a right angle at the vertex O. We need to find the locus of the midpoint of PQ and then determine the latus rectum of the resulting conic.
Step 2: Key Formula or Approach:
For the parabola \( y^2 = 4ax \), use parametric coordinates for points on the curve: \( (at^2, 2at) \).
The condition that OP is perpendicular to OQ means the product of their slopes is -1.
Let the midpoint of PQ be \( (h, k) \). Express \( h \) and \( k \) in terms of the parameters of P and Q, and eliminate the parameters to find the locus.
Step 3: Detailed Explanation:
The given parabola is \( y^2 = 4x \), so \( a = 1 \).
The vertex is \( O(0,0) \).
Let the coordinates of P be \( (t_1^2, 2t_1) \) and Q be \( (t_2^2, 2t_2) \).
Slope of OP \( = m_{OP} = \frac{2t_1 - 0}{t_1^2 - 0} = \frac{2}{t_1} \).
Slope of OQ \( = m_{OQ} = \frac{2t_2 - 0}{t_2^2 - 0} = \frac{2}{t_2} \).
Since OP and OQ are perpendicular, \( m_{OP} \times m_{OQ} = -1 \).
\( \left(\frac{2}{t_1}\right) \left(\frac{2}{t_2}\right) = -1 \implies \frac{4}{t_1 t_2} = -1 \implies t_1 t_2 = -4 \).
Let \( M(h, k) \) be the midpoint of the line segment PQ.
By the midpoint formula:
\( h = \frac{t_1^2 + t_2^2}{2} \implies t_1^2 + t_2^2 = 2h \).
\( k = \frac{2t_1 + 2t_2}{2} \implies t_1 + t_2 = k \).
We need to eliminate \( t_1 \) and \( t_2 \) to find the relation between \( h \) and \( k \).
We know the algebraic identity: \( (t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \).
Substitute the values we've established:
\( k^2 = 2h + 2(-4) \).
\( k^2 = 2h - 8 \).
\( k^2 = 2(h - 4) \).
To find the locus, replace \( (h, k) \) with \( (x, y) \):
\( y^2 = 2(x - 4) \).
This is an equation of a parabola of the form \( Y^2 = 4A X \), where \( Y = y \), \( X = x - 4 \), and \( 4A = 2 \).
The conic C is a parabola. The length of its latus rectum is the coefficient of the linear term, which is \( 4A \).
Therefore, the length of the latus rectum is 2.
Step 4: Final Answer:
The length of the latus rectum is 2.