CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


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.
loader
B
Assertion is true, Reason is true and Reason is NOT correct explanation for Assertion.
loader
C
Assertion is true, Reason is false
loader
D
Assertion is false, Reason is true
loader

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

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image