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What does the Circle revision sheet cover?

A Circle is a Locus, The Three Forms, standard, centre-radius, diameter, ends A(x_1,y_1), B(x_2,y_2), General Equation of a Circle, general form, centre and radius, Parametric Form, standard circle x^2+y^2=r^2, centre (h,k), Tangent to a Circle.

Is this for Class 11 Maharashtra HSC Mathematics?

Yes, it is written for the Maharashtra HSC Mathematics syllabus.

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Circle — Class 11 Maharashtra HSC Mathematics revision notes

A 4-page visual revision sheet for Circle, written for Class 11 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.

Circle revision notes, page 1 of 4 — A preview of page 1 — all 4 pages open full size in the app.

Circle in one look

every point the same distance from one centre, in three equivalent equations and one tangent formula

A Circle is a Locus

a circle is the set of all points in a plane equidistant from one fixed point
that fixed point is the centre, that fixed distance is the radius
the standard form x^2+y^2=r^2 is the circle centred at the origin with radius r
the centre-radius form (x-h)^2+(y-k)^2=r^2 is the same circle recentred at (h,k)
it's built from CP=r and the distance formula
the diameter form (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0 comes from the angle-in-a-semicircle theorem
if A,B are the ends of a diameter and P is any point on the circle
then ∠ APB=90°, so AP⊥ BP
the product of their slopes is -1

The Three Forms

standard x^2+y^2=r^2
centre-radius (x-h)^2+(y-k)^2=r^2
diameter, ends A(x_1,y_1), B(x_2,y_2) (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0

Q. Find the equation of the circle with A(2,-3) and B(-3,5) as ends of a diameter

diameter form: (x-2)(x+3)+(y+3)(y-5)=0
expand: x^2+x-6+y^2-2y-15=0

Answer: x^2+y^2+x-2y-21=0

Q. Centre (3,-4), and the line 3x-4y-5=0 cuts the circle in a chord of length 6. Find the equation.

the perpendicular distance from the centre to the chord: CM=frac|3(3)-4(-4)-5|√3^2+4^2=|9+16-5|/5=20/5=4
half the chord AM=3, so by Pythagoras on right triangle AMC: CA^2=CM^2+AM^2=16+9=25, giving radius CA=5
centre-radius form: (x-3)^2+(y+4)^2=25

Answer: x^2+y^2-6x+8y=0

Page 1 of 4. The full sheet, all 4 pages, opens free in the app.

What this Circle sheet covers

A Circle is a Locus
The Three Forms
standard
centre-radius
diameter, ends A(x_1,y_1), B(x_2,y_2)
General Equation of a Circle
general form
centre and radius
Parametric Form
standard circle x^2+y^2=r^2
centre (h,k)
Tangent to a Circle
at (x_1,y_1) on x^2+y^2=r^2
at (x_1,y_1) on x^2+y^2+2gx+2fy+c=0
condition of tangency for y=mx+c on x^2+y^2=a^2
Tangents from an External Point, and the Director Circle
Director Circle
of x^2+y^2=a^2
Circles touching externally
Circles touching internally
Disjoint circles
Intersecting circles
Traps that cost marks
Tricks that save time

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