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Standard XII

Physics

Power and Thin Lenses in Contact

Consider the ...

Question

Consider the conic $e x^{2} + π y^{2} - 2 e^{2} x - 2 π^{2} y + e^{3} + π^{3} = π e$ . Suppose $P$ is any point on the conic and $S_{1}, S_{2}$ are the foci of the conic, then the maximum value of $(P S_{1} + P S_{2})$ is

π e

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\sqrt{π e}

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2 \sqrt{π}

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2 \sqrt{e}

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Solution

The correct option is C $2 \sqrt{π}$
$e x^{2} + π y^{2} + - 2 e^{2} x - 2 π^{2} y + e^{3} + π^{3} = π e e (x^{2} - 2 e x + e^{2}) + π (y^{2} - 2 π y + π^{2}) = π e \frac{{(x - e)}^{2}}{π} + \frac{{(y - π)}^{2}}{e} = 1$
$a = \sqrt{π}, b = \sqrt{e}$
for ellipse sum of focal distance $P S + P S^{'}$ is $2 a$
So $P S + P S^{'} = 2 \sqrt{π}$
option $C$ is correct

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......

(*)

On the basis of above information answer the following question:

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(*)

If the focus is

(α, β)

& the directrix is

a x + b y + c = 0

then the equation of conic whose eccentricity

= e

is given by

{(x - α)}^{2} + {(y - β)}^{2} = e^{2} \frac{{(a x + b y + c)}^{2}}{a^{2} + b^{2}}

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e = 1

then conic is called parabola, for

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Now consider the conic

169 {{(x - 1)}^{2} + {(y - 3)}^{2}} = {(5 x - 12 y + 17)}^{2}

......

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On the basis of above information answer the following question:

The equation of directrix of the conic

(*)

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Consider the conic ex2+πy2−2e2x−2π2y+e3+π3=πe. Suppose P is any point on the conic and S1,S2 are the foci of the conic, then the maximum value of (PS1+PS2) is

The correct option is C 2√πex2+πy2+−2e2x−2π2y+e3+π3=πee(x2−2ex+e2)+π(y2−2πy+π2)=πe(x−e)2π+(y−π)2e=1a=√π,b=√efor ellipse sum of focal distance PS+PS′ is 2aSo PS+PS′=2√πoption C is correct

Consider the conic $e x^{2} + π y^{2} - 2 e^{2} x - 2 π^{2} y + e^{3} + π^{3} = π e$ . Suppose $P$ is any point on the conic and $S_{1}, S_{2}$ are the foci of the conic, then the maximum value of $(P S_{1} + P S_{2})$ is