Question:
The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:
The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:
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For chord length problems in parabolas, use intersection substitution and standard chord length formulas.
Updated On: May 19, 2025
- \( 16a\sqrt{5} \)
- \( 8a\sqrt{5} \)
- \( 4a\sqrt{5} \)
- \( 2a\sqrt{5} \)
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The Correct Option is A
Solution and Explanation
Step 1: Finding the intersection points
We substitute \( x = \frac{y + 3}{2} \) into the parabola equation \( y^2 = 4ax \):
\[
y^2 = 4a \left( \frac{y + 3}{2} \right)
\]
\[
y^2 - 2ay - 6a = 0
\]
Solving this quadratic equation in \( y \) gives the points \( P(y_1) \) and \( Q(y_2) \).
Step 2: Finding the distance PQ
Using the chord length formula:
\[
PQ = \frac{|2a|}{\sqrt{1 + (m^2)}}
\]
where \( m = 2 \) (slope of line),
\[
PQ = \frac{2a \times \sqrt{5}}{1}
\]
\[
PQ = 16a\sqrt{5}
\]
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