MHT-CET Maths · Circle

Concentric Circles and Circles Touching a Line or an Axis

A concentric circle keeps −g and −f and changes only c; a circle touches a line when its radius equals the distance from the centre to that line, and it touches the X-axis when r = |k|.

All Circle notes MHT-CET Maths strategy

Why this matters

10 PYQs at 30% HARD — one of the chapter's smallest pages and its most repetitive. Concentric with a given circle and double its area (twice), concentric and passing through the centre of another circle, concentric and touching the X-axis, centre given and touching a line, and the point of contact of a circle with a line. Every one is 'keep the centre, find r' or 'r is a distance'.

Concept 1 of 2: Concentric Circles: Same Centre, New Radius From Area or From a Point

Concentric circles share

g

and

f

; only the constant changes. Double the area means

R^2 = 2r^2

; passing through a given point means

R

is the distance from the centre to that point.

Definition

$x^2 + y^2 - 6x - 4y - 12 = 0$ : centre $(3, 2)$ , $r = 5$ . Double area: $R^2 = 50$ : $(x - 3)^2 + (y - 2)^2 = 50 \Rightarrow x^2 + y^2 - 6x - 4y = 37$ .
$2x^2 + 2y^2 - 6x + 8y + 1 = 0$ : scale to $x^2 + y^2 - 3x + 4y + \tfrac12 = 0$ , $r^2 = \tfrac94 + 4 - \tfrac12 = \tfrac{23}{4}$ ; double: $R^2 = \tfrac{23}{2}$ ; new constant $\tfrac{25}{4} - \tfrac{23}{2} = -\tfrac{21}{4}$ : $4x^2 + 4y^2 - 12x + 16y - 21 = 0$ .
Concentric with $2x^2 + 2y^2 - 8x - 12y - 9 = 0$ (centre $(2, 3)$ ) and through the centre $(-4, -5)$ of another circle: $R = \sqrt{36 + 64} = 10$ : $x^2 + y^2 - 4x - 6y - 87 = 0$ .
Scale the leading coefficient to $1$ BEFORE reading $g, f, c$ ; a factor of $2$ left in place halves the centre wrongly.

Concentric family

x^2 + y^2 + 2gx + 2fy + \lambda = 0 \qquad \text{double area: } R^2 = 2r^2

Worked example

Find the circle concentric with

x^2 + y^2 + 4x - 2y - 4 = 0

and passing through

(1, 5)

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 3rd May Shift 1 · Q127Moderate

Example 1 · Circle · Concentric Circles and Circles Touching a Line or an Axis

The equation of the concentric circle, with circle

C_1

having equation

x^2+y^2-6x-4y-12=0

and having double the area of

C_1

, is

Doubling the radius for double the area

Double AREA means

R = \sqrt2\, r

R^2 = 2r^2

. Doubling

r

quadruples the area and gives

x^2 + y^2 - 6x - 4y = 87

, which is not on the list — but

50

(the value of

R^2

, not the constant) is.

Drill 2 more on concentric circles: same centre, new radius from area or from a point

Concept 2 of 2: Touching a Line or an Axis: Radius = Distance From the Centre; Contact Point = Foot of the Perpendicular

A line is tangent when its distance from the centre equals the radius. So a circle with a known centre that touches a given line has

r =

that distance, and the point of contact is the foot of the perpendicular from the centre. Touching the

X

-axis means

r = |k|

, the

Y

-axis

r = |h|

Definition

Centre $(3, 4)$ , touching $5x + 12y - 11 = 0$ : $r = \dfrac{|15 + 48 - 11|}{13} = 4$ : $x^2 + y^2 - 6x - 8y + 9 = 0$ .
Concentric with centre $(3, 2)$ and touching the $X$ -axis: $r = 2$ : $x^2 + y^2 - 6x - 4y + 9 = 0$ .
Centre $(-1, 1)$ touching $x + 2y + 4 = 0$ : the contact point $(h, k)$ is on the line and $CP$ is perpendicular to it (slope $2$ ): $h + 2k = -4$ , $2h - k = -3$ : $(-2, -1)$ .
Touching both axes means $|h| = |k| = r$ ; in the first quadrant the centre is $(r, r)$ .

Tangency of a line

\frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} = r \qquad \text{touches the } X\text{-axis} \iff r = |k|

Worked example

Find the circle with centre

(2, -3)

that touches the line

3x - 4y + 7 = 0

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 4th May Shift 1 · Q136Moderate

Example 2 · Circle · Concentric Circles and Circles Touching a Line or an Axis

The equation of the circle which has its centre at the point

(3,4)

and touches the line

5x+12y-11=0

Using the centre's x-coordinate for tangency to the X-axis

Touching the

X

-axis fixes

r = |k|

, the

y

-coordinate. With centre

(3, 2)

r = 2

, constant

9

; using

r = 3

gives constant

4

, option (D).

Drill 2 more on touching a line or an axis: radius = distance from the centre; contact point = foot of the perpendicular

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Concentric Circles: Same Centre, New Radius From Area or From a Point
Concentric family
$x^2 + y^2 + 2gx + 2fy + \lambda = 0 \qquad \text{double area: } R^2 = 2r^2$
Touching a Line or an Axis: Radius = Distance From the Centre; Contact Point = Foot of the Perpendicular
Tangency of a line
$\frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} = r \qquad \text{touches the } X\text{-axis} \iff r = |k|$

Watch out for (2)

Doubling the radius for double the area
→ Concentric Circles: Same Centre, New Radius From Area or From a Point
Using the centre's x-coordinate for tangency to the X-axis
→ Touching a Line or an Axis: Radius = Distance From the Centre; Contact Point = Foot of the Perpendicular

Test yourself on Circle

15 past MHT-CET questions from this chapter, timed at 27 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circle chapter test Drill all 10 questions

Related notes

Straight Line — the foot of the perpendicular →