Question

which one of the following statements are correct?
1) Equation of a circle with centre $(h,k)$ and radius r is given by $(x-h{)}^2+(y-k{)}^2=r^2$
2) Equation of circle with $A(x_1,y_1)$ and $B(x_2,y_2)$ as extremities of diameter is given by

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

Answer 1

Let C(h,k) be the centre of the circle and P(x,y) be any point on the circle.

PC=radius=r

$\Rightarrow \sqrt{(x-h{)}^2+(y-k{)}^2=r}$

$\Rightarrow (x-h{)}^2+(y-k{)}^2=r^2$

Any point on the circle satisfies this

$\Rightarrow$ statement 1 is correct

AB is a diameter and P is a general point on the circle.

$\angle APB={90}^{\circ }$

$\Rightarrow$ slope of $PA\times slopeofPB=-1$

$\Rightarrow \frac{y-y_1}{x-x_1}\times \frac{y-y_2}{x-x_2}=-1$

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

Any point on the circle satsfies this

$\Rightarrow$ statement 2 is wrong.