JEE Main Syllabus 2025 PDF (Physics, Chemistry, Maths)

# Family of Circles - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Family of Circles is considered one of the most asked concept.

• 86 Questions around this concept.

## Solve by difficulty

If the minimum radius of the circle which contains the three circles

$\begin{gathered} x^2-y^2-4 y-5=0 \\ x^2+y^2+12 x+4y+31=0 \end{gathered}$

$\text { and } x^2+y^2+6 x+12y+36=0 \: i\! s\: \left(3+\frac{5}{36} \sqrt{\lambda}\right)$
Then the value of  $\lambda$ must be

A square is inscribed in the circle $x^2+y^2-10 x-6 y+30=0$. One side of the square is parallel to $y=x+3$, then one vertex of the square is

What is the equation of the circle that passes through the points $(1,1),(2,3), \text { and }(-1,2) ?$

Let the ellipse E : \begin{aligned} & x^2+9y^2=9 \\ \end{aligned}  intersect the positive x-and y-axes at the points A and B respectively. Let the
major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the
point P. If the area of the triangle with vertices A, P and the origin O is  \begin{aligned} & \frac{m}{n} \end{aligned}  where m and n are coprime, then m – n is equal to :

Equation of the circle touching the circle $\mathrm{x^2+y^2-15 x+5 y=0}$ at the point $\mathrm{(1,2)}$ and passing through the point $\mathrm{(0,2)}$ is

The equation of the circle having its centre on the line $\mathrm{x+2 y-3=0}$ and passing through the point of intersection of the circle $\mathrm{x^{2}+y^{2}-2 x-4 y+1=0}$ and $\mathrm{x^{2}+y^{2}-4 x-2 y+4=0}$ is

If the distances from the origin to the centres of three circles $x^2+y^2+2 \lambda_i x-c^2=0(i=1,2,3)$ are in G.P. then the lengths of the tangents drawn to them from any point on the circle $x^2+y^2=c^2$ are in

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Tangents are drawn from any point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$. If the chord of contact touches the circle $x^2+y^2=c^2, a>b$, then

The equation of the circle through the points of intersection of $\mathrm{x^2+y^2-1=0, x^2+y^2-2 x-4 y+1=0}$ and touching the line $\mathrm{x+2 y=0}$ is:

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The equations of the three circles are: $\mathrm{ x^2+y^2-12 x-16 y+64=0, \quad 3 x^2+3 y^2-36 x+81=0 }$ and

$\mathrm{x^2+y^2-16 x+81=0}$. Find the coordinates of the point from which the length of tangents drawn to each of the three circles is equal.

## Concepts Covered - 1

Family of Circles

Family of Circles

1.     Equation of the family of circles passing through the point of intersection of two given circles S = 0 and S’ = 0 is S + λS’ = 0 where, λ is the parameter

2.     Equation of the family of circles passing through the point of intersection of a given circle S = 0 and a line L = 0is S + λL = 0 where, λ is the parameter.

3.     Equation of the family of circles touching the given circle  S = 0 and the line L = 0 is S + λL=0

4.     Equation of the family of circles passing through the two given points P(x1, y1)  and Q(x2, y2) is

$\left(\mathrm{x}-\mathrm{x}_{1}\right)\left(\mathrm{x}-\mathrm{x}_{2}\right)+\left(\mathrm{y}-\mathrm{y}_{1}\right)\left(\mathrm{y}-\mathrm{y}_{2}\right)+\lambda\left|\begin{array}{lll}{x} & {y} & {1} \\ {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1}\end{array}\right|=0$

5.     Equation of the family of circles which touch $\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right) \text { at }\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ for any finite m is  $\left(\mathrm{x}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}-\mathrm{y}_{1}\right)^{2}+\lambda\left\{\left(\mathrm{y}-\mathrm{y}_{1}\right)-\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right)\right\}=0$

And if m is infinite then the family of circles is $\left(\mathrm{x}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}-\mathrm{y}_{1}\right)^{2}+\lambda\left(\mathrm{x}-\mathrm{x}_{1}\right)=0$

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Family of Circles

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## Books

### Reference Books

#### Family of Circles

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 4.33

Line : 52