The locus of the vertices of the family of parabolas y = 3 a 3 x 2 + 2 a 2 x − 2 a is

Parabola: y = 3 a 3 x 2 + 2 a 2 x − 2 a Vertex: ( α , β ) α = 2 a 3 /3 − a 2 /2 = − 4 a 3 , β = 4 3 a 3 − ( 4 a 4 + 4 ⋅ 3 a 3 ⋅ 2 a ) = − 3 4 a 3 − ( 4 1 + 3 8 ) a 4 = − 12 35 4 a × 3 = − 16 35 a α β = − 4 a 3 ( − 16 35 ) a = 64 105 .

Mathematics - Parabola Question with Solution | TestHub

MathematicsParabolaGeneralHard2 minPYQ_2006

MathematicsHardsingle choice

The locus of the vertices of the family of parabolas $y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2 a$ is

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Parabola: $y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2 a$ Vertex: $(α, β)$ $α = \frac{- a ^{2} /2}{2 a ^{3} /3} = - \frac{3}{4 a}, β = \frac{- ( \frac{a ^{4}}{4} + 4 \cdot \frac{a ^{3}}{3} \cdot 2 a )}{4 \frac{a ^{3}}{3}} = - \frac{- ( \frac{1}{4} + \frac{8}{3} ) a ^{4}}{\frac{4}{3} a ^{3}} = - \frac{35}{12} \frac{a}{4} \times 3 = - \frac{35}{16} a α β = - \frac{3}{4 a} (- \frac{35}{16}) a = \frac{105}{64} .$

Stream:JEESubject:MathematicsTopic:ParabolaSubtopic:General

⏱ 2mℹ️ Source: PYQ_2006

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