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The tangents at the extremities of a focal chord of a parabola
A. are perpendicular
B. are parallel
C. intersect on the directrix
D. intersect at the vertex
Options:
A. are perpendicular
B. are parallel
C. intersect on the directrix
D. intersect at the vertex
Solution:
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Verified Answer
The correct answers are:
intersect on the directrix, are perpendicular
Let the tangent at $\mathrm{P}$ (at ${ }^2, 2 \mathrm{at}$ ) on the parabola is $t y=x+a t^2$ and at the other end $\left(\frac{a}{t^2}, \frac{-2 a}{t}\right)$ of the focal chord through $\mathrm{P}$
$-\frac{1}{t} y=x+\frac{a}{t^2} \Rightarrow y=-t x-\frac{a}{t}$
Product of the slopes $=\frac{1}{t}(-t)=-1$ and the point of intersection lies on $\left(\frac{1}{t}+\mathrm{t}\right) \mathrm{x}+\mathrm{a}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)=0 \Rightarrow \mathrm{x}=-\mathrm{a}$, the directrix
$-\frac{1}{t} y=x+\frac{a}{t^2} \Rightarrow y=-t x-\frac{a}{t}$
Product of the slopes $=\frac{1}{t}(-t)=-1$ and the point of intersection lies on $\left(\frac{1}{t}+\mathrm{t}\right) \mathrm{x}+\mathrm{a}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)=0 \Rightarrow \mathrm{x}=-\mathrm{a}$, the directrix
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