Circle Formulas – HSC Math

Posted In: Mathematics
1. With centre $(\h,\k)$ and radius $\r$ a circle is:
$(\x-\h)^2 + (\y-\k)^2 = \r^2$
2. If centre $(\h,\k)$ and passes through $(\α,\β)$:
$(\x-\h)^2 + (\y-\k)^2 = (\α-\h)^2 + (\β-\k)^2$
3. General equation of a circle:
$\x^2+\y^2+2\gx+2\fy+\c=0$
where $(-\g,-\f)$ is centre and radius, $\r=\g^\2+\f^\2−\c$. (g, f and c are three constants)
4. Position of $(\x_1,\y_1)$ with respect to a circle:
i. Outside of the circle, if $\x_1^2+\y_1^2+2\gx_1+2\hy_1+c$ > 0
ii. Upon the circle, if $\x_1^2+\y_1^2+2\gx_1+2\hy_1+c$ = 0
iii. Inside the circle, if $\x_1^2+\y_1^2+2\gx_1+2\hy_1+c$ < 0
5. Intercept on the axis:
on the y-axis = $2√{\f^2-\c}=2√{\r^2-\h^2}$
on the x-axis = $2√{\g^2-\c}=2√{\r^2-\k^2}$
if the circle touches the y-axis then $\c=\f^2$
if the circle touches the x-axis then $\c=\g^2$
6. A circle which has diameter between $(\x_\1,\y_\1)$ and $(\x_\2,\y_\2)$:
$(\x-\x_\1)(\x-\x_\2)+(\y-\y_\1)(\y-\y_\2)=\0$
7. A circle passes through intersection of a circle and a straight line:
$\x^\2+\y^\2+\2fy+\k(\lx+\my+\n)=0$
where,
intersection circle, $\x^\2+\y^\2+\2\gx+\2\fy+\c=0$
intersection straight line, $\lx+\my+\n=\0$
8. A circle passes through intersection of two other circles:
$\first\_\circle+\k(\second\_\circle)=0$
where k is a constant.
9. A circle passing through any two points $(\x_1,\y_1)$ and $(\x_2,\y_2)$:
$(\x-\x_1)(\x-\x_2)+(\y-\y_1)(\y-\y_2)+\k((\x-\x_1)(\y_1-\y_2)-(\y-\y_1)(\x_1-\x_2))=0$
[Law of AR Khalifa]
10. When two circles touch each other:
External touch: $\C_1\C_2 = \r_1+\r_2$
Internal touch: $\C_1\C_2 = |\r_1-\r_2|$
11. The equation of a circle in polar co-ordinates with center at $(\r_\1,\θ_\1)$ and radius $\a$ is:
$\a^2=\r^2+\r_1^2+2\rr_1\cos(\θ-\θ_1)$
12. Condition for tangency of a straight line:
$c=±√{\m^2+1}$
where the line is $\y=\mx+\c$ and the circle is $\x^2+\y^2=\r^2$
13. Length of a tangent drawn from an external point $(\x_1,\y_1)$
=$√{\x_1^2+\y_1^2+2\gx_1+2\fy_1+\c}$
14. Chord of a circle having middle point at $(\x_1,\y_1)$ is:
$\xx_1+\yy_1+\g(\x+\x_1)+\f(\y+\y_1)+\c=\x_1^2+\y_1^2+2\gx_1+2\fy_1+\c$
$[T=S_1]$
15. Normal at the point $(\x_1,\y_1)$ of a circle:
$\x_1\y-\xy_1=0$
16. Equation of the chord of contact of tangents drawn from an external points $(\x_1,\y_1)$ to a circle:
$\xx_1+\yy_1+\g(\x+\x_1)+\f(\y+\y_1)+c=0$
17. Equation of common chord of two circles:
$\S_1-\S_2=0$
i.e. $\first\_\circle-\second\_\circle=0$
18. Touch point of a circle and a tangent:
$({{-\mr}/√{1+\m^2},{\r/√{1+\m^2})$
where the circle is $\x^2+\y^2=\r^2$ and tangent is $\y=\mx+\c$.

Post Tags: Geometry | HSC