Length of the focal chord of the parabola ( y + 3 ) 2 = − 8 ( x − 1 ) which lies at a distance 2 units from the vertex of the parabola is

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Lengths are invariant under change of axes

consider $y^{2} = 8 x$ . Consider focal chord at $(2 t^{2}, 4 t)$

Focus $= (2, 0)$ . Equation of focal chord at is $y = \frac{2 t}{t ^{2} - 1} 9 x - 2) \Rightarrow 2 t x + (1 - t^{2}) y - 4 t = 0$ $\frac{4∣ t ∣ ^{2}}{4 t ^{2} + ( 1 - t ^{2} ) ^{2}} = 2 \Rightarrow (∣ t ∣ - 1)^{2} = 0$

Length of focal chord at ' t ' $= 2 (t + \frac{1}{t})^{2} = \frac{2 ( t ^{2} + 1 ) ^{2}}{t ^{2}} = 8$

Mathematics - Parabola Question with Solution | TestHub

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