MHT-CET · Chapter

Circle

The concepts MHT-CET needs, one connected idea at a time.

0 / 9

concepts mastered

Ready for you

How the ideas build on each other. Each one opens once you've mastered what it needs.

the signs inside are opposite to the centre's

Start

Diameter formEnds of a diameter

A(x_1, y_1)

,

B(x_2, y_2)

:

(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0

Locked

The general equation

x^2 + y^2 + 2gx + 2fy + c = 0

: centre

(-g, -f)

, radius

\sqrt{g^2 + f^2 - c}

Locked

Finding a circle from conditionsThrough three points, touching an axis, centre on a line: each condition is one equation in g, f, cLocked

Parametric form

x = h + r\cos\theta

,

y = k + r\sin\theta

traces the circle once as

\theta

goes roundLocked

Tangent at a pointReplace $x^2$ by $xx_1$ , $2x$ by $x + x_1$ (and the same for $y$ ): $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$ Locked
Condition of tangency $y = mx + c$ touches $x^2 + y^2 = a^2$
Tangents from a point; director circleLength of tangent $= \sqrt{S_{11}}$ ;

Two circles and their common tangentsCompare the distance between the centres with $r_1 + r_2$ and $|r_1 - r_2|$ : it decides touching, cutting and the number of common tangentsLocked

when

c^2 = a^2(1 + m^2)

:

the distance from the centre equals

r

Locked

perpendicular tangents to

x^2 + y^2 = a^2

meet on

x^2 + y^2 = 2a^2

Locked