X2-y2-a2 and x-c2-y2-b2 are two intersecting circles- lf a- b- c are the sides BC- CA- AB of -ABC- lf p1-p2-p3 are the altitudes through A-B-C respectively then the length of the common chord is

The correct option is C 2p_3 When two circles intersect at 2 different points the #10;common chord and the line joining the center are perpendicular bise ...

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Standard XII

Mathematics

Length of Chord of Contact

x2+y2=a2 and ...

Question

$x^{2} + y^{2} = a^{2}$ and $(x - c)^{2} + y^{2} = b^{2}$ are two intersecting circles. lf $a, b, c$ are the sides $B C, C A, A B$ of $Δ A B C$ . lf $p_{1}, p_{2}, p_{3}$ are the altitudes through $A, B, C$ respectively then the length of the common chord is

2 p_{1}

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2 p_{2}

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2 p_{3}

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p_{1}

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Solution

The correct option is C $2 p_{3}$
When two circles intersect at 2 different points the common chord and the line joining the center are perpendicular bisectors.

$O C = O D = \frac{C D}{2}$

$⟹ C D = 2 \times O C$

$A B ⊥ C D$

$⟹ O C is the altitude of the triangle and it passes through C .$

$⟹ O C = P_{3} ⟹ C D = 2 P_{3}$

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Similar questions

Q. If in a triangle

A B C, a, b, c

are in A.P. and

p_{1}, p_{2}, p_{3}

are the altitudes from the vertices

A, B, C

respectively then

Q. If

p_{1}, p_{2}, p_{3}

are respectively the lengths of perpendiculars from the vertices of a triangle

A B C

to the opposite sides then
(i)

\frac{cos A}{p_{1}} + \frac{cos B}{p_{2}} + \frac{cos C}{p_{3}} = \frac{1}{R}

(ii)

\frac{b p_{1}}{c} + \frac{c p_{2}}{a} + \frac{a p_{3}}{b} = \frac{a^{2} + b^{2} + c^{2}}{2 R}

Q. If in a

△ A B C, a, b, c

are in A.P. and

p_{1}, p_{2}, p_{3}

are the altitudes from the vertices

A, B, C

respectively, then

Q. If in a triangle

A B C, a, b, c

are in A.P. and

p_{1}, p_{2}, p_{3}

are the altitudes from the vertices

A, B, C

respectively then

Q. The chord of contact of tangents drawn from any point on the circle

x^{2} + y^{2} = a^{2}

to the circle

x^{2} + y^{2} = b^{2}

touches

x^{2} + y^{2} = c^{2}

, where

a, b, c > 0,

then

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are in

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x2+y2=a2 and (x−c)2+y2=b2 are two intersecting circles. lf a, b, c are the sides BC, CA, AB of ΔABC. lf p1,p2,p3 are the altitudes through A,B,C respectively then the length of the common chord is

The correct option is C 2p3When two circles intersect at 2 different points the common chord and the line joining the center are perpendicular bisectors. OC=OD=CD2 ⟹CD=2×OC AB⊥CD ⟹OC is the altitude of the triangle and it passes through C. ⟹OC=P3⟹CD=2P3

$x^{2} + y^{2} = a^{2}$ and $(x - c)^{2} + y^{2} = b^{2}$ are two intersecting circles. lf $a, b, c$ are the sides $B C, C A, A B$ of $Δ A B C$ . lf $p_{1}, p_{2}, p_{3}$ are the altitudes through $A, B, C$ respectively then the length of the common chord is