Question

# Assertion :The chord of contact of tangent from a point $$P$$ to a circle passes through $$Q$$. If $${ l }_{ 1 }$$ and $${ l }_{ 2 }$$ are the lengths of the tangents from $$P$$ and $$Q$$ to the circle, then $$PQ$$ is equal to $$\sqrt { { { l }_{ 1 } }^{ 2 }+{ { l }_{ 2 } }^{ 2 } }$$ Reason: The equation of chord of contact of tangents from the point $$P\left( { x }_{ 1 },{ y }_{ 1 } \right)$$ to the circle $${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$$ is $$x{ x }_{ 1 }+y{ y }_{ 1 }={ a }^{ 2 }$$

A
Assertion is true, Reason is true and reason is correct explanation for Statement-1.
B
Assertion is true, Reason is true and Reason is NOT correct explanation for Assertion.
C
Assertion is true, Reason is false
D
Assertion is false, Reason is true

Solution

## The correct option is A Assertion is true, Reason is true and reason is correct explanation for Statement-1.Let $$P\equiv \left( { x }_{ 1 },{ y }_{ 1 } \right)$$ and $$Q\equiv \left( { x }_{ 2 },{ y }_{ 2 } \right)$$Let the equation of the given circle be $${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$$The equation of chord of contact of tangents from the point $$P\left( { x }_{ 1 },{ y }_{ 1 } \right)$$ to the given circle is $$x{ x }_{ 1 }+y{ y }_{ 1 }={ a }^{ 2 }$$Since it passes through $$Q\left( { x }_{ 2 },{ y }_{ 2 } \right)$$$$\therefore { x }_{ 1 }{ x }_{ 2 }+{ y }_{ 1 }{ y }_{ 2 }={ a }^{ 2 }$$    ...(1)Now $${ l }_{ 1 }=\sqrt { { { x }_{ 1 } }^{ 2 }+{ { y }_{ 1 } }^{ 2 }-{ a }^{ 2 } } ,{ l }_{ 2 }=\sqrt { { { x }_{ 2 } }^{ 2 }+{ { y }_{ 2 } }^{ 2 }-{ a }^{ 2 } }$$and $$PQ=\sqrt { { \left( { x }_{ 2 }-{ x }_{ 1 } \right) }^{ 2 }+{ \left( { y }_{ 2 }-{ y }_{ 1 } \right) }^{ 2 } }$$$$=\sqrt { \left( { { x }_{ 1 } }^{ 2 }+{ { y }_{ 1 } }^{ 2 } \right) +\left( { { x }_{ 2 } }^{ 2 }+{ { y }_{ 2 } }^{ 2 } \right) -2\left( { x }_{ 1 }{ x }_{ 2 }+{ y }_{ 1 }{ y }_{ 2 } \right) } \\ =\sqrt { \left( { { x }_{ 1 } }^{ 2 }+{ { y }_{ 1 } }^{ 2 } \right) +\left( { { x }_{ 2 } }^{ 2 }+{ { y }_{ 2 } }^{ 2 } \right) -2{ a }^{ 2 } } \\ =\sqrt { \left( { { x }_{ 1 } }^{ 2 }+{ { y }_{ 1 } }^{ 2 }-{ a }^{ 2 } \right) +\left( { { x }_{ 2 } }^{ 2 }+{ { y }_{ 2 } }^{ 2 }-{ a }^{ 2 } \right) } =\sqrt { { { l }_{ 1 } }^{ 2 }+{ { l }_{ 2 } }^{ 2 } }$$Maths

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