Question:
If the chords of the hyperbola $x^2 - y^2 = a^2$ touch the parabola $y^2 = -4ax$, then the locus of the middle points of these chords is
If the chords of the hyperbola $x^2 - y^2 = a^2$ touch the parabola $y^2 = -4ax$, then the locus of the middle points of these chords is
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Use discriminant = 0 for tangency problems involving curves.
Updated On: Apr 23, 2026
- $y^2 = (x-a)x^3$
- $y^2(x-a) = x^3$
- $x^2(x-a) = x^3$
- None of these
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The Correct Option is D
Solution and Explanation
Concept:
Chord with given midpoint + tangency condition
Step 1: Use midpoint form of chord.
For hyperbola: \[ xx_1 - yy_1 = a^2 \]
Step 2: Apply tangency condition with parabola.
Substitute in $y^2 = -4ax$ and use discriminant = 0.
Step 3: Eliminate parameter.
Simplification leads to equation not matching any given options. Conclusion:
Answer = None of these
Step 1: Use midpoint form of chord.
For hyperbola: \[ xx_1 - yy_1 = a^2 \]
Step 2: Apply tangency condition with parabola.
Substitute in $y^2 = -4ax$ and use discriminant = 0.
Step 3: Eliminate parameter.
Simplification leads to equation not matching any given options. Conclusion:
Answer = None of these
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