First slide

Parabola

Question

Locus of the mid points of the chords of the parabola
$y^{2} = 8 x$ which touch the circle $x^{2} + y^{2} = 4$ is

Moderate

A

${(y^{2} - 4 x)}^{2} = 16 (x^{2} + 4)$
B

$(y^{2} - 4 x^{2}) = 4 (x^{2} + a^{2})$
C

${(y^{2} - 4 x)}^{2} = 4 (16 + y^{2})$
D

${(y^{2} - 4 x)}^{2} = 16 (4 + y^{2})$

Solution

Let ( $h, k$ ) be the mid-point of the chord of the
parabola $y^{2} = 8 x$ then its equation is
$\begin{array}{l} k y - 4 (x + h) = k^{2} - 8 h \\ \Rightarrow 4 x - k y + k^{2} - 4 h = 0 \end{array}$
which touches the circle $x^{2} + y^{2} = 4$
$\begin{array}{l} \Rightarrow \frac{k^{2} - 4 h}{\sqrt{16 + k^{2}}} = 2 \\ \Rightarrow {(k^{2} - 4 h)}^{2} = 4 (k^{2} + 16) \end{array}$
Locus of $(h, k$ ) is ${(y^{2} - 4 x)}^{2} = 4 (16 + y^{2})$ .

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