Let centre of a circle, passing through point (0,0),(1,0) and touching circle x^2+y^2=9 , be (h, k)

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10-1.Circle and System of Circles

hard

Let the centre of a circle, passing through the point $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to .............

$1$

$2$

$6$

$9$

(JEE MAIN-2024)

Solution

$Image$

$ (\mathrm{x}-\mathrm{h})^2+(\mathrm{y}-\mathrm{k})^2=\mathrm{h}^2+\mathrm{k}^2 $

$ \mathrm{x}^2+\mathrm{y}^2-2 \mathrm{hx}-2 \mathrm{ky}=0 $

$ \because \text { passes through }(1,0) $

$ \Rightarrow 1+0-2 \mathrm{~h}=0 $

$ \Rightarrow \mathrm{h}=1 / 2 $

$ \because \mathrm{OC}=\frac{\mathrm{OP}}{2} $

$ \sqrt{\left(\frac{1}{2}\right)^2+\mathrm{k}^2}=\frac{3}{2}$

$ \frac{1}{4}+\mathrm{k}^2=\frac{9}{4} $

$ \mathrm{k}^2=2 $

$ \mathrm{k}= \pm \sqrt{2}$

$\therefore$ Possible coordinate of

$ \mathrm{c}(\mathrm{h}, \mathrm{k})\left(\frac{1}{2}, \sqrt{2}\right)\left(\frac{1}{2},-\sqrt{2}\right) $

$ 4\left(\mathrm{~h}^2+\mathrm{k}^2\right)=4\left(\frac{1}{4}+2\right)=4\left(\frac{9}{4}\right)=9$

Standard 11

Mathematics

Let the latus ractum of the parabola $y ^{2}=4 x$ be the common chord to the circles $C _{1}$ and $C _{2}$ each of them having radius $2 \sqrt{5}$. Then, the distance between the centres of the circles $C _{1}$ and $C _{2}$ is

medium

(JEE MAIN-2020)

The equation of a circle passing through origin and co-axial to circles ${x^2} + {y^2} = {a^2}$ and ${x^2} + {y^2} + 2ax = 2{a^2},$ is

hard

The equation of the circle having its centre on the line $x + 2y – 3 = 0$ and passing through the points of intersection of the circles ${x^2} + {y^2} – 2x – 4y + 1 = 0$ and ${x^2} + {y^2} – 4x – 2y + 4 = 0$, is

hard

The number of common tangents, to the circles $x^2+y^2-18 x-15 y+131=0$ and $x^2+y^2-6 x-6 y-7=0$, is :

medium

(JEE MAIN-2023)

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hard

Let the centre of a circle, passing through the point $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to .............

Solution

Similar Questions

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