Good morning class, today we will start with new chapter, Circles. Everybody knows what is a circle.

Circle is locus of a point which moves in such a way that its distance from a fixed point remains

constant. So the locus of that point which moves in such a way that from a fixed point the distance is

always constant. This constant distance is called as radius. And the fixed point is called centre. So

everybody of you knows what is a circle.

Let us talk about equations of circle. Equation of circle whose centre is x1, y1 and radius is a, is x

minus x1 whole square plus y minus y1 whole square equals to a square, centre x1y1 radius a.

Equation of centre of circle is origin and radius is a is x square plus y square is equal to a square. If

you put 0,0 in the place of x1y1 you will get x square plus y square is equal to a square.

Solve one question, find equation of circle with centre 1,2 and passing through point minus 2,6. Find

equation of circle with centre 1,2 passing through point minus 2,6. Hence radius will be this distance,

by distance formula we can find radius is 3 square plus 4 square under the root that is 5. Hence

equation of circle is x minus 1 whole square, x minus x1 whole square plus y minus y1 whole square

is equal to r square. Solving this we get x square plus y square minus 2 x minus 4y minus 20 equals to

0, solving this we get this equation, this is equation of circle. So to find the equation of circle we

always will need two things, centre and radius, if we get the centre and radius then the equation of

circle is very easy to get. x minus x1 whole square plus y minus y1 whole square equals to a square.

General equation of circle, x square plus y square plus 2gx plus 2fy plus c is equal to zero represents

general equation of circle. In this the coefficient of x square and y square is the same and the

coefficient of xy is zero. So in general 2 degree equation, when the coefficient of x2 and y2 is the

same and the coefficient of xy is zero, the equation will be the equation of circle. If I complete the

whole square then we can see this equation becomes x plus g whole square plus y plus f whole

square equals g square plus f square minus c. From here we can see that centre is minus g, minus f

and radius is square root of g square plus f square minus c. Centre is minus g, minus f, radius is

square root of g square plus f square minus c.

If we take a small example, see this is the equation. This will be the equation of a circle. The centre

will be half of coefficient of x, 2, change the sign, half of coefficient of y, 3, change the sign of that

also, and radius will be square root of g square plus f square minus c. Minus and minus is plus 1 that

is square root of 14. Clear? Centre is minus g, minus f and radius is square root of g square plus f

square minus c.

Let us start solving few questions. Find equation of circle which passes through centre of this circle, x

square plus y square plus 8 x plus 10 y minus 7 equal to zero. Centre is minus 4, minus 5. And is

concentric with this circle. While simplifying this circle we will divide it by 2, now this will come into a

standard format. The centre of this will be minus g, minus f that is minus 2, 3. We have to look out

for such a circle which passes through the first point and the second point will be its centre. Hence

find equation of circle passing through centre of first circle and concentric with this circle. Hence this

will be radius, so what is radius, minus 6 whole square, 8 square under root 100 that is 10. So

equation of circle is, equation of this required circle is x minus x1 whole square plus y minus y1

whole square equals to r square. On simplifying we will get x square plus y square minus 4x minus 6

y plus 4 plus 9 plus 13 minus 87 equals to zero. x square plus y square minus 4 x minus 6y minus 87 is

equal to zero. Is this clear to everybody? Very good.

Let us solve next question. Equation of circle whose radius is 5, centre lies on y axis and which

passes through 3,2. Radius is 5, centre lies on y axis, it is on y axis so the y coordinate will be zero.

Let us assume centre to be 0,a. It is passing through 3,2, radius is 5. We will make the equation,

make the radius 5, hence 9 plus a minus 2 whole square is equal to 25, a minus 2 whole square is

equal to 16, a minus 2 is 4, or a minus 2 is minus 4, a equals to 6, or a equals to minus 2. There are

two values of 1 so there are corresponding two circles. First will be x….