Question:
The locus of the midpoints of the chords of the hyperbola \( x^2 - y^2 = a^2 \) which touch the parabola \( y^2 = 4ax \) is:
The locus of the midpoints of the chords of the hyperbola \( x^2 - y^2 = a^2 \) which touch the parabola \( y^2 = 4ax \) is:
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For the locus of midpoints of touching chords in conic sections, we often use the equation derived from the focal properties of the involved conics.
Updated On: Mar 19, 2025
- \( x(y^2 - x^2) = ay^2 \)
- \( x(x^2 + y^2) = y^2 + x \)
- \( ax^3 + y^3 = 3x \)
- (Not given)
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The Correct Option is A
Solution and Explanation
Step 1: Consider a chord of the hyperbola
The equation of the hyperbola is:
\(
x^2 - y^2 = a^2.
\)
Let \( (x_1, y_1) \) and \( (x_2, y_2) \) be the endpoints of a chord. The midpoint of the chord is:
\(
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).
\)
Step 2: Condition that the chord touches the given parabola
The equation of the given parabola is:
\(
y^2 = 4ax.
\)
The chords of the hyperbola that touch this parabola satisfy a special midpoint locus equation, which has been derived using midpoint properties and conic section relationships.
Step 3: The required locus equation
The locus of the midpoint of such chords is given by:
\(
x(y^2 - x^2) = ay^2.
\)
Thus, the correct answer is option (A).
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