Question

If \( O \) is the vertex of the parabola \( x^2 = 4ay \), \( Q \) is the point on the parabola. If \( C \) is the locus of the point which divides \( OQ \) in ratio 2:3, the equation of the chord of \( C \) which is bisected at point \( (1, 2) \) is

• A. \( 5x + 4y + 3 = 0 \)
• B. \( 5x - 4y - 3 = 0 \)
• C. \( 5x - 4y + 3 = 0 \)
• D. \( 5x + 4y - 3 = 0 \)

Hint:

When finding the equation of a chord, use the section formula and the midpoint to simplify and find the equation.

Answer 1

The Correct Option is C

Step 1: Use the equation of the parabola. 
The equation of the parabola is: \[ x^2 = 4ay \] where \( a \) is the focal length. The point \( Q \) lies on the parabola, and the coordinates of \( Q \) are \( (x, y) \).

 Step 2: Find the coordinates of point \( C \). 
The point \( C \) divides the line segment \( OQ \) in the ratio 2:3. Using the section formula, we find the coordinates of \( C \). The section formula gives the point dividing the line in the ratio \( m:n \) as: \[ C = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] 

Step 3: Apply the section formula. 
We apply this formula to find the coordinates of point \( C \) and substitute the values. 

Step 4: Find the equation of the chord. 
Using the mid-point formula and simplifying, we obtain the equation of the chord of the parabola as: \[ 5x - 4y + 3 = 0 \]