Question:

Consider the hyperbola x2100y264=1\frac{x^2}{100}-\frac{y^2}{64}=1 with foci at SS and S1S_1, where SS lies on the positive xx-axis Let PP be a point on the hyperbola, in the first quadrant Let SPS1=α\angle \operatorname{SPS}_1=\alpha, with α<π2\alpha<\frac{\pi}{2}. The straight line passing through the point SS and having the same slope as that of the tangent at PP to the hyperbola, intersects the straight line S1PS _1 P at P1P _1 Let δ\delta be the distance of PP from the straight line SP1SP _1, and β=S1P\beta= S _1 P. Then the greatest integer less than or equal to βδ9sinα2\frac{\beta \delta}{9} \sin \frac{\alpha}{2} is _____.

Updated On: May 20, 2024
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The answer is 7.

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Consider the hyperbola x2/100-y2/64=1 with foci at S and S1GivenSPPP=20We have βδsinα2=20Squaring the above equation and rearranging terms, we obtain β2+δ2sin2α2400=2βδsinα2The reciprocal of SP is related to δ as 1SP=sinα2δcosα=SP2+β26562βθsinσ2=2βδsin22562βSsinα2=cosαSolving for λλ128λ=cosαλ(1cosα)=128βδsinα22sin2α2=128βδ9sinα2=649[βδ9sinα2]=7 where [.] denotes greatest integer function \begin{aligned} &Given\\& S P-P P=20 \\ & We\ have\ \beta-\frac{\delta}{\sin \frac{\alpha}{2}}=20 \\ &Squaring\ the\ above\ equation\ and\ rearranging\ terms,\ we\ obtain  \\ & \beta^2+\frac{\delta^2}{\sin ^2 \frac{\alpha}{2}}-400=\frac{2 \beta \delta}{\sin \frac{\alpha}{2}} \\ &The\ reciprocal\ of\ SP\ is\ related\ to\ \delta\ as\ \frac{1}{S P}=\frac{\sin \frac{\alpha}{2}}{\delta} \\ & \cos \alpha=\frac{S P^2+\beta^2-656}{2 \beta \frac{\theta}{\sin \frac{\sigma}{2}}} \\ & =\frac{\frac{2 \beta \delta}{\frac{\sin }{2}}-256}{\frac{2 \beta S}{\sin \frac{\alpha}{2}}}=\cos \alpha \\ & Solving\ for\ \lambda\\ & \frac{\lambda-128}{\lambda}=\cos \alpha \\ & \lambda(1-\cos \alpha)=128 \\ & \frac{\beta \delta}{\sin \frac{\alpha}{2}} \cdot 2 \sin ^2 \frac{\alpha}{2}=128 \\ & \frac{\beta \delta}{9} \sin \frac{\alpha}{2}=\frac{64}{9}\\ &\Rightarrow\left[\frac{\beta \delta}{9} \sin \frac{\alpha}{2}\right]=7 \text { where [.] denotes greatest integer function } \\ & \end{aligned}

So, the answer is 7.

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Concepts Used:

Hyperbola

Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.

Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.

Hyperbola