If P S Q and P S prime R are two chords of an ellipse through its foci S and S respectively, then (P S/S Q)+(P S prime/S prime R)=

Question

If P S Q and P S prime R are two chords of an ellipse through its foci S and S respectively, then (P S/S Q)+(P S prime/S prime R)= (A) 2((1+e2/1-e2)) (B) (1+e2/1-e2) (C) 2((1-e2/1+e2)) (D) (1-e2/1+e2)

Tardigrade Admin · Accepted Answer

We know that semi latus rectum is harmonic mean of the segments of focal chord.

∴ \frac{1}{SP} + \frac{1}{SQ} = \frac{2 a}{b ^{2}}

... (i)
and

\frac{1}{S ^{'} P} + \frac{1}{S ^{'} R} = \frac{2 a}{b ^{2}}

.... (ii)
Also

SP + S^{'} P = 2 a

.... (iii)
From (i),

1 + \frac{SP}{SQ} = \frac{2 a}{b ^{2}} SP

From (ii),

1 + \frac{S ^{'} P}{S ^{'} R} = \frac{2 a}{b ^{2}} S^{'} P

On adding, we get

2 + \frac{SP}{SQ} + \frac{S ^{'} P}{S ^{'} R} = \frac{2 a}{b ^{2}} (SP + S^{'} P)

\Rightarrow 2 + \frac{SP}{SQ} + \frac{S ^{'} P}{S ^{'} R} = \frac{4 a ^{2}}{b ^{2}} = \frac{4}{1 - e ^{2}}

\Rightarrow \frac{SP}{SQ} + \frac{S ^{'} P}{S ^{'} R} = \frac{4}{1 - e ^{2}} - 2 = 2 \frac{1 + e ^{2}}{1 - e ^{2}}

Q. If $PSQ$ and $P S^{'} R$ are two chords of an ellipse through its foci $S$ and $S$ respectively, then $\frac{PS}{SQ} + \frac{P S ^{'}}{S ^{'} R} =$

$2 (\frac{1 + e ^{2}}{1 - e ^{2}})$

$\frac{1 + e ^{2}}{1 - e ^{2}}$

$2 (\frac{1 - e ^{2}}{1 + e ^{2}})$

$\frac{1 - e ^{2}}{1 + e ^{2}}$

Q. If PSQ and PS′R are two chords of an ellipse through its foci S and S respectively, then SQPS​+S′RPS′​=

2(1−e21+e2​)

1−e21+e2​

2(1+e21−e2​)

1+e21−e2​

Q. If $PSQ$ and $P S^{'} R$ are two chords of an ellipse through its foci $S$ and $S$ respectively, then $\frac{PS}{SQ} + \frac{P S ^{'}}{S ^{'} R} =$

$2 (\frac{1 + e ^{2}}{1 - e ^{2}})$

$\frac{1 + e ^{2}}{1 - e ^{2}}$

$2 (\frac{1 - e ^{2}}{1 + e ^{2}})$

$\frac{1 - e ^{2}}{1 + e ^{2}}$