7. The distance from the center of circle $C : x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$ to the line $3 x + 4 y + 4 = 0$ is $d =$ $\_\_\_\_$.

5. The distance from the center of circle $C : x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$ to the line $l : 3 x + 4 y + 4 = 0$ is $d =$ $\_\_\_\_$ $3$. Analysis: This examines the point-to-line distance formula. The distance from the center $(1,2)$ to the line $3 x + 4 y + 4 = 0$ is $\frac { | 3 \times 1 + 4 \times 2 + 4 | } { 5 } = 3$

If a circle passing through point $(2,1)$ is tangent to both coordinate axes, then the distance from the center of the circle to the line $2 x - y - 3 = 0$ is
A．$\frac { \sqrt { 5 } } { 5 }$
B．$\frac { 2 \sqrt { 5 } } { 5 }$
C．$\frac { 3 \sqrt { 5 } } { 5 }$
D．$\frac { 4 \sqrt { 5 } } { 5 }$

5. The distance from the point $( 3,0 )$ to an asymptote of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ is
A. $\frac { 9 } { 5 }$
B. $\frac { 8 } { 5 }$
C. $\frac { 6 } { 5 }$
D. $\frac { 4 } { 5 }$

Find the distance from the center of the circle $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y = 0$ to the line $x - y + 2 = 0$

154. Suppose lines $x + y = 1$ and $x - y = 3$ are the diameters of a circle, and the line $4x + 3y + 5 = 0$ is tangent to it. What is the distance of point $M(4, -2)$ from the circle?

• [(1)] $\sqrt{3} - 1$
• [(2)] $\sqrt{3} - \sqrt{2}$
• [(3)] $\dfrac{\sqrt{7}}{2}$
• [(4)] $\sqrt{5} - 2$

Let the lines $( 2 - i ) z = ( 2 + i ) \bar { z }$ and $( 2 + i ) z + ( i - 2 ) \bar { z } - 4 i = 0$, (here $i ^ { 2 } = - 1$ ) be normal to a circle $C$. If the line $i z + \bar { z } + 1 + i = 0$ is tangent to this circle $C$, then its radius is :
(1) $\frac { 3 } { \sqrt { 2 } }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { 2 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

Let a circle $C$ of radius 5 lie below the $x$-axis. The line $L _ { 1 } : 4 x + 3 y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L _ { 2 } : 3 x - 4 y - 11 = 0$ at $Q$. The line $L _ { 2 }$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x - 12 y + 51 = 0$ is

Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$

Q84. Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$

Let the planes $\pi _ { 1 } \equiv x + y = 1$ and $\pi _ { 2 } \equiv x + z = 1$.\ a) ( 1.5 points) Find the planes parallel to plane $\pi _ { 1 }$ such that their distance to the origin of coordinates is 2.\ b) ( 0.5 points) Find the line that passes through the point ( $0,2,0$ ) and is perpendicular to plane $\pi _ { 2 }$.\ c) ( 0.5 points) Find the distance between the points of intersection of plane $\pi _ { 1 }$ with the $x$ and $y$ axes.

The point $P$ has coordinates $( p , q )$, and the equation of a circle is
$$x ^ { 2 } + 2 f x + y ^ { 2 } + 2 g y + h = 0$$
where $f , g , h , p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$. Which one of the following is sufficient on its own to be able to calculate $L$ ?
A the values of $f , g$ and $h$
B the values of $f , g , p$ and $q$
C the values of $f , h , p$ and $q$
D the values of $g , h , p$ and $q$
E none of the options A-D is sufficient on its own

A circle drawn in the rectangular coordinate plane

• has one common point with the line $d_{1}: y - \frac{4x}{3} - 46 = 0$,
• has two common points with the line $d_{2}: y - \frac{4x}{3} - 6 = 0$,
• has no common points with the line $d_{3}: y - \frac{4x}{3} - 1 \geqslant 0$.

It is known that. Accordingly, which of the following could be the radius of this circle in units?
A) 11 B) 13 C) 15 D) 17 E) 19