Circle

The point on the parabola $y^2=4 a x$ nearest to the focus has its abscissa

Let the straight line $\mathrm{y}=2 \mathrm{x}$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $\mathrm{r}$ at a point $\mathrm{A}_1$. Let $\mathrm{B}_1$ be the point on the circle such that the line…

Let $G$ be a circle of radius $R>0$. Let $G_{1},G_{2},\ldots,G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1},G_{2},\ldots,G_{n}$ touches the circle $G$ externally. Also, for…

Let the circles $C_{1}:x^{2}+y^{2}=9$ and $C_{2}:(x-3)^{2}+(y-4)^{2}=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_{3}:(x-h)^{2}+(y-k)^{2}=r^{2}$ satisfies the following conditions: $(i)$…

Let the circles $C_{1}:x^{2}+y^{2}=9$ and $C_{2}:(x-3)^{2}+(y-4)^{2}=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_{3}:(x-h)^{2}+(y-k)^{2}=r^{2}$ satisfies the following conditions: $(i)$…

A line $y=mx+1$ intersects the circle $(x-3)^{2}+(y+2)^{2}=25$ at the points $P$ and $Q.$ If the midpoint of the line segment $PQ$ has $x-$coordinate $-\frac{3}{5},$ then which one of the following options is correct?

Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_{1}$ be the set of all pairs of circles $(S_{1},S_{2})$ such that $T$ is tangents to $S_{1}$ at $P$ and tangent to $S_{2}$ at $Q$, and also…

Paragraph: Let $S$ be the circle in the $xy$ - plane defined by the equation $x^{2}+y^{2}=4.$ Question : Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect…

Paragraph: Let $S$ be the circle in the $xy$ - plane defined by the equation $x^{2}+y^{2}=4.$ Question : Let $E_{1}E_{2}$ and $F_{1}F_{2}$ be the chords of $S$ passing through the point $P_{0}(1,1)$ and parallel to the…

The circle $C_{1}:x^{2}+y^{2}=3,$ with centre at O, intersects the parabola $x^{2}=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_{1}$ at P touches other two circles $C_{2}$ and $C_{3}$ at…

Let RS be the diameter of the circle $x^{2}+y^{2}=1$ where, $S$ is the point$(1,0)$. Let $P$be a variable point (other than $R\&S$) on the circle and tangents to the circle at $S\&P$meet at the point$Q$. The normal to…

A tangent $P T$ is drawn to the circle $x^{2}+y^{2}=4$ at the point $P(\sqrt{3}, 1)$. A straight line $L$, perpendicular to $P T$ is a tangent to the circle $(x-3)^{2}+y^{2}-1$. Question: A common tangent of the two…