If centre of a circle which passing through points of intersection of circles {x^2} + {y^2} - 6x + 2

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10-1.Circle and System of Circles

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If the centre of a circle which passing through the points of intersection of the circles ${x^2} + {y^2} - 6x + 2y + 4 = 0$and ${x^2} + {y^2} + 2x - 4y - 6 = 0$ is on the line $y = x$, then the equation of the circle is

A

$7{x^2} + 7{y^2} - 10x + 10y - 11 = 0$

B

$7{x^2} + 7{y^2} + 10x - 10y - 12 = 0$

C

$7{x^2} + 7{y^2} - 10x - 10y - 12 = 0$

D

$7{x^2} + 7{y^2} - 10x - 12 = 0$

Solution

(c) Family of circles through points of intersection of two circles is ${S_1} + \lambda {S_2},\,\,(\lambda \ne – 1)$.

${x^2} + {y^2} – 6x + 2y + 4 + \lambda ({x^2} + {y^2} + 2x – 4y – 6) = 0$

Centre is $(3 – \lambda ,\; – 1 + 2\lambda )$.

It lies on $y = x$.

Therefore, $ – 1 + 2\lambda = 3 – \lambda $

$\Rightarrow \lambda = \frac{4}{3}$

Hence equation of circle can be found by substituting $\lambda $ in the family of circles above.

Standard 11

Mathematics

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Similar Questions

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let the circles $C_1: x^2+y^2=9$ and $C_2:(x-3)^2+(y-4)^2=16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3:(x-h)^2+(y-k)^2=r^2$ satisfies the following conditions :

$(i)$ centre of $C _3$ is collinear with the centres of $C _1$ and $C _2$

$(ii)$ $C _1$ and $C _2$ both lie inside $C _3$, and

$(iii)$ $C _3$ touches $C _1$ at $M$ and $C _2$ at $N$.

Let the line through $X$ and $Y$ intersect $C _3$ at $Z$ and $W$, and let a common tangent of $C _1$ and $C _3$ be a tangent to the parabola $x^2=8 \alpha y$.

There are some expression given in the $List-I$ whose values are given in $List-II$ below:

$List-I$ $List-II$

$(I)$ $2 h + k$ $(P)$ $6$

$(II)$ $\frac{\text { Length of } ZW }{\text { Length of } XY }$ $(Q)$ $\sqrt{6}$

$(III)$ $\frac{\text { Area of triangle } MZN }{\text { Area of triangle ZMW }}$ $(R)$ $\frac{5}{4}$

$(IV)$ $\alpha$ $(S)$ $\frac{21}{5}$

$(T)$ $2 \sqrt{6}$

$(U)$ $\frac{10}{3}$

($1$) Which of the following is the only INCORRECT combination?

$(1) (IV), (S)$ $(2) (IV), (U)$ $(3) (III), (R)$ $(4) (I), (P)$

($2$) Which of the following is the only CORRECT combination?

$(1) (II), (T)$ $(2) (I), (S)$ $(3) (I), (U)$ $(4) (II), (Q)$

Give the answer or quetion ($1$) and ($2$)

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