1
Question
Two conics have a common focus; prove that two of their common chords pass through the intersection of their directrices.
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Solution
Take the equation to the conics as
l1r=−1−e1cosθ....1
and l2r=−1−e2cosθ....2
Their directrices are respectively,
rcosθ=l1e1.....3
and rcos(θ−α)=−l2e2.....4
To get the equation to the common chords of 1 and 2, subtract them
Subtracting, we get
l1r−l2r=e1cosθ−e2cos(θ−α)....5
Clearly 5 passes through that point.
Similarly, we can prove for the other common chord.
l1r=−1−e1cosθ....1
and l2r=−1−e2cosθ....2
Their directrices are respectively,
rcosθ=l1e1.....3
and rcos(θ−α)=−l2e2.....4
To get the equation to the common chords of 1 and 2, subtract them
Subtracting, we get
l1r−l2r=e1cosθ−e2cos(θ−α)....5
Clearly 5 passes through that point.
Similarly, we can prove for the other common chord.
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