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Line and Circle is considered one the most difficult concept.
124 Questions around this concept.
The equation of the circle of radius which touches the line is:
Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line $\mathrm{X}+\mathrm{Y}=1$. If the intercepts made by the circle $\mathrm{x}^2+\mathrm{y}^2-\mathrm{x}+3 \mathrm{y}=0$ on $L_1$ and $L_2$ are equal, then which of the following equations can represent $L_1$:
If touches the circle , then prove that the point lies.
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Image of the circle in the line is
Let a variable line passing through the centre of the circle $x^2+y^2-16 x-4 y=0$, meet the positive co-ordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$. Then the minimum value of $\mathrm{OA}+\mathrm{OB}$, where $\mathrm{O}$ is the origin is equal to
If the straight line $y=m x$ is outside the circle $x^2+y^2-20 y+90=0$, then
Line and Circle
$S$ is a circle with center $O$ and radius $r$, and $L$ is a straight line in the plane of the circle.
Equation of circle $S: x^2+y^2=a^2$
Equation of line $L: y=m x+c$
To find their point(s) of intersection, we can solve these equations simultaneously
$
\begin{aligned}
& x^2+(m x+c)^2=a^2 \\
& \left(1+m^2\right) x^2+2 m c x+c^2-a^2=0
\end{aligned}
$
Case (1)
If line L intersects the circle S in two distinct points, then Equation (iii) will have two real and distinct roots
So, Discriminant of equation (iii) > 0
In this case the line is a secant to the circle, i.e., it represents the equation of chord to the circle.
Case (2)
If the line $L$ touches the circle, then Equation (iii) will have two equal real roots
So, Discriminant of equation (iii) $=0$
$
\begin{aligned}
& \mathrm{B}^2-4 \mathrm{AC}=0 \\
& 4 \mathrm{~m}^2 \mathrm{c}^2-4\left(1+\mathrm{m}^2\right)\left(\mathrm{c}^2-\mathrm{a}^2\right)=0 \\
& \mathrm{a}^2=\frac{\mathrm{c}^2}{1+\mathrm{m}^2} \\
& c^2=a^2\left(1+m^2\right)
\end{aligned}
$
In this case the line is a tangent to the circle
This is also the condition of tangency to the circle.
Case (3)
If the line L and the circle S have no common points, then Equation (iii) will have imaginary roots
So, Discriminant of equation (iii) < 0
Alternate method to check position of line with respect to the circle
We can find the distance OM between the line and the centre of the circle, and compare it with radius of the circle
If $\mathrm{OM}<\mathrm{r}$, then line is a chord (secant) to the circle
If $\mathrm{OM}=r$, then line is a tangent to the circle
If $\mathrm{OM}>\mathrm{r}$, then line does not intersect the circle at any point
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