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# If P and Q Are Two Points Whose Coordinates Are - CBSE Class 10 - Mathematics

#### Question

If P and Q are two points whose coordinates are (at2 ,2at) and (a/t2 , 2a/t) respectively and S is the point (a, 0). Show that \frac{1}{SP}+\frac{1}{SQ} is independent of t.

#### Solution

We have,

SP=sqrt((at^2-a)^2+(2at-0)^2)

=sqrt((t^2-1)^2+4t^2)=a(t^2+1)

=>SQ=sqrt((a-a/t^2)^2+(0+(2a)/t)^2)

=>SQ=sqrt((a^2(1-t^2))/t^4+(4a^2)/t^2)

=>SQ=a/t^2sqrt((1-t^2)^2+4t^2)=a/t^2sqrt((1+t^2)^2)

which is independent of t.

=\frac{a}{t^{2}}(1 +\t^{2})

\therefore\frac{1}{SP}+\frac{1}{SQ}=1/(a(t^2+1))+t^2/(a(t^2+1)

\Rightarrow\frac{1}{SP}+\frac{1}{SQ}=(1+t^2)/(a(t^2+1)) = 1/a

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Solution If P and Q Are Two Points Whose Coordinates Are Concept: Distance Formula.
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