A set of parallel chords of the parabola y 2-4 ax have their mid points onA- any straight line through the vertexB- any straight line through the focusC- a straight line parallel to the axisD- another parabola

The correct option is C a straight line parallel to the axis Let points P(at^2_1, 2at_1) and Q(at^2_2, 2at_2) lie on the parabola y^2=4ax Here, the points P an ...

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Standard XII

Mathematics

Director Circle

A set of para...

Question

A set of parallel chords of the parabola $y^{2} = 4 a x$ have their mid points on

any straight line through the vertex

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any straight line through the focus

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a straight line parallel to the axis

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another parabola

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Solution

The correct option is C
a straight line parallel to the axis

Let points $P (a t_{1}^{2}, 2 a t_{1})$ and $Q (a t_{2}^{2}, 2 a t_{2})$ lie on the parabola $y^{2} = 4 a x$
Here, the points P and Q are variable but slope of the chord PQ, $m_{P Q} = \frac{2}{t_{1} + t_{2}} .$
Now let the mid point of PQ be R(h, k),
$k = \frac{2 a t_{1} + 2 a t_{2}}{2} o r k = a (t_{1} + t_{2}) = \frac{2}{m} \Rightarrow y = \frac{2}{m}$
This is a line parallel to the axis of the parabola

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A set of parallel chords of the parabola y2=4ax have their mid points on

A set of parallel chords of the parabola $y^{2} = 4 a x$ have their mid points on