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Byju's Answer

Standard XII

Mathematics

Position of a Line with Respect to Circle

Let a circle ...

Question

Let a circle $C$ touch the lines $L_{1} : 4 x - 3 y + K_{1} = 0$ and $L_{2} : 4 x - 3 y + K_{2} = 0, K_{1}, K_{2} \in R$ . If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(- 1, 2)$ and $L_{2}$ at $(3, - 6)$ , then the equation of the circle $C$ is:

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Solution

Co-ordinate of centre $C \equiv (\frac{3 + (- 1)}{2}, \frac{- 6 + 2}{2}) \equiv (1, - 2)$
$L_{1}$ is passing through $A$
$\Rightarrow - 4 - 6 + K_{1} = 0$
$\Rightarrow K_{1} = 10$
$L_{2}$ is passing through $B$
$\Rightarrow 12 + 18 + K_{2} = 0$
$\Rightarrow K_{2} = - 30$
Equation of $L_{1} : 4 x - 3 y + 10 = 0$
Equation of $L_{2} : 4 x - 3 y - 30 = 0$
Diameter of circle $= ∣ ∣ ∣ ∣ ∣ \frac{10 + 30}{\sqrt{4^{2} + (- 3)^{2}}} ∣ ∣ ∣ ∣ ∣ = 8$
$\Rightarrow$ Radius $= 4$
Equation of circle ${(x - 1)}^{2} + {(y + 2)}^{2} = 16$

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Similar questions

Q. Consider

L_{1} : 2 x + 3 y + p - 3 = 0

L_{2} : 2 x + 3 y + p + 3 = 0

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where

p

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C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0

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STATEMENT 1 : If line

L_{1}

is a chord of circle

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L_{2}

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STATEMENT 2 : If line

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Q. Assertion :Consider

L_{1} : 2 x + 3 y + p - 3 = 0 L_{2} : 2 x + 3 y + p + 3 = 0,

where p is a real number, and

C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0

Statement-1: If line

L_{1}

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Reason: If line

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Q. Assertion :Consider

L_{1} : 2 x + 3 y + p - 3 = 0

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where p is a real number and

C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0

If the line

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L_{2}

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Reason: If the line

L_{1}

is a diameter of the circle C then the line

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Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line x + y = 1. If the intercepts made by

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Let a circle C touch the lines L1:4x−3y+K1=0 and L2:4x−3y+K2=0,K1,K2∈R. If a line passing through the centre of the circle C intersects L1 at (−1,2) and L2 at (3,−6), then the equation of the circle C is:

Let a circle $C$ touch the lines $L_{1} : 4 x - 3 y + K_{1} = 0$ and $L_{2} : 4 x - 3 y + K_{2} = 0, K_{1}, K_{2} \in R$ . If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(- 1, 2)$ and $L_{2}$ at $(3, - 6)$ , then the equation of the circle $C$ is: