ParabolaHardBloom L3
Question
For the parabola $y^{2} + 4x - 4y = 4$, the straight line $x - y + 3 = 0$ is :
Options
A.focal chord
B.normal chord
C.both focal chord and normal chord
D.none of these
Solution
Sol. [Key Concept: Reduce to standard form $(y{- 2)}^{2} = - 4(x - 2)$; then test whether the line is a focal chord and/or a normal chord.]
$$(y - 2)^{2} = - 4(x - 2)$$
$$\begin{matrix} y - 2 & \ = Y,x - 2 = X \\ Y^{2} & \ = - 4X \end{matrix}$$
Line is $X - Y = - 3$
$$\begin{array}{r} \Rightarrow \ Y\ = X + 3\#(1) \\ Y\ = - tX - 2t - t^{3}\#(2) \end{array}$$
(1) and (2) are identical for $t = - 1$
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