The locus of the mid-points of the chords of the hyperbola x²-y²=4, which touch the parabola y²=8 x, is [26 Aug 2021 Shift 2]

Correct Answer is : y2(x−2)=x3y²(x-2)=x³y2(x−2)=x3

Correct Answer is : y2(x−2)=x3y²(x-2)=x³y2(x−2)=x3

Test Index

Hyperbola

Section: Mathematics

Question : 59 of 98

Marks: +1, -0

x^{2}-y^{2}=4

y^{2}=8 x

Solution:

Let the mid-point of the chord is

(h_{1} k)

. Then, chord through mid - point

(h, k)

T=S_{1}

x h-y k=h^{2}-k^{2}

...(i)

Now, this is also a tangent of

y^{2}=8 x

The equation of the tangent of slope

m

to the parabola

y^{2}=8 x

is given by

Tangent :

y=m x+\frac{2}{m}

\Rightarrow m^{2} x-m y=-2

...(ii)

Eqs. (i) and (ii) are coincide

\therefore \frac{h}{m^{2}}=\frac{-k}{-m}=\frac{h^{2}-k^{2}}{-2}

\Rightarrow h=k m

\Rightarrow m=\frac{h}{k}

\therefore \frac{k^{2}}{h}=\frac{h^{2}-k^{2}}{-2}

\Rightarrow -2 k^{2}=h^{3}-h k^{2}

\Rightarrow h^{3}=k^{2}(h-2)

Therefore, locus of mid-point of the chords,

x^{3}=y^{2}(x-2)

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