Math topic中文

Midpoint

midpoint of line segment

Practice questions

1In $\Delta ABC$ , $AC = 12\text{ cm}$ , $\angle ABC$ is a right angle, $\angle ACB = \frac{\pi}{6}$ radians and $M$ is the…
2A circle with centre O passes through the points P(-1,7) and Q(0,8). State the relationship between the perpendicular bi…
3Two points $A(-3, -6)$ and $B(7, -6)$ lie on a circle with centre $G$ . The equation of the tangent to the circle at the…
4A(1, 7) and B(9, 7) lie on a circle $C_1$ with centre P. The line $4y + 3x = 31$ is a normal to the circle and passes th…
5 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
6Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
7Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows…
8Solutions to this question by accurate drawing will not be accepted. The diagram shows a trapezium $ABCD$ in which $AB$ …
9已知过点 $P(1,\dfrac{3}{2})$ 的曲线 $C$ 的方程为 $\sqrt{{{(x-1)}^{2}}+{{y}^{2}}}+\sqrt{{{(x+1)}^{2}}+{{y}^{2}}}=2a.(Ⅰ)$ 求曲线 $C$ 的标准…
10In the diagram, $OPQR$ is a trapezium with $OP$ parallel to $RQ$ and $OP$ perpendicular to $PQ$ . The point $P$ has coord…
11The points $P$ and $Q$ both lie on a circle and have coordinates $(-10, 15)$ and $(10, 0)$ respectively. The centre of t…
12点 $(a,b)$ 关于直线 $x+y+1=0$ 的对称点是___.
13The lines $y=9$ , $y=-1$ and $3y=4x+9$ are tangents to a circle. The x-coordinate of the centre of the circle is positive…
14The line $y + 2x = 12$ intersects the curve $y = 2x^2 - 6x - 4$ at two distinct points $A$ and $B$ . Find the coordinates…
15已知 $A(2,-3)$ ， $\overrightarrow{AB}=(3,-2)$ ，则线段 $AB$ 的中点坐标为___.
16直角坐标平面上有一个轴对称图形，点A（3，-1）、B（3，-7）是此图形上的一对对称点．若此图形上有一点C（-2，-9），则点C在图形上的一个对称点坐标为．
17The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
18The line $y = x - 4$ and the curve $y = \frac{2}{x - 3}$ intersect at the points $A$ and $B$ . Find the length of $A B$ .
19The figure shows three points $A$ , $C$ and $D$ whose coordinates are $(3, 5)$ , $(4, -2)$ and $(7, 2)$ respectively. Find…
20Solutions to this question by accurate drawing will NOT be accepted. The figure shows a right-angled triangle $ABC$ ,…
21The diagram shows a quadrilateral $ABCD$ with vertices $A(-4, -3)$ , $B(6, 1)$ , $C(3, 5)$ and $D(-2, 3)$ . $M$ which lies…
22A researcher conducted a Particle Filtration Efficiency (PFE) test on surgical masks and reusable masks which have been…
23The diagram shows points $A(5, 0)$ , $B(a, 3a)$ and $C(-2, 6)$ such that the line $AB$ is perpendicular to the line $BC$ .…
24点 $A(3,1)$ 关于点 $M(4,-3)$ 的对称点是 $B$ ，直线 $l$ 过原点且过 $B$ 点，则直线 $l$ 方程为；
25In the diagram below, the line $OP$ makes an angle of $\alpha^\circ$ with the positive $x$ -axis such that $\tan \alpha = 0.2$ …
26Solutions to this question by accurate drawing will not be accepted. $OABC$ is a trapezium with right angle $OAB$ . T…
27The points $A(5, 7)$ and $B(-11, 15)$ lie on a circle $C$ . The centre of the circle $O$ lies on the line $y = -2x - 3$ .…
28The equation of a circle, $C_1$ is $x^2 + y^2 - 4x + 6y - 3 = 0$ . Find the coordinates of the centre and the radius of $C_1$ …
29已知双曲线 $x^{2}- \dfrac {y^{2}}{3}=1$ 上存在两点 $M$ ， $N$ 关于直线 $y=x+m$ 对称，且 $MN$ 的中点在抛物线 $y^{2}=18x$ 上，则实数 $m$ 的值为 ___ ．
30过双曲线 $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1(a>0,b>0)$ 的右焦点 $F$ 作一条渐近线的垂线，垂足为 $M$ ，且 $FM$ 的中点 $A$ 在双曲线上，则双曲线离心率 $e$ …
31The diagram (not drawn to scale) shows a quadrilateral $ABCD$ such that $AB = BC$ and angle $BCD = 90^\circ$ . The point…
32The straight line $2x + y = 1$ intersects the curve $x^2 - xy - y^2 + 5 = 0$ at the points $A$ and $B$ . Show that the le…
33The diagram below is not drawn to scale. In the diagram, D, E and F are points on the circle C1. Show that DE is the dia…
34The equation of a circle with centre C is $x^{2}+y^{2}-8x-2y-152=0$ . Find the coordinates of C and the radius of the cir…
35The diagram shows three points $P(-6, 0)$ , $Q(4k, k-2)$ and $R(8, 5)$ where $PQ$ is perpendicular to $QR$ and $k < 0. Show that$ …
36The diagram shows a trapezium PQRS in which PQ is parallel to SR. The point P is (-1, -2) and the point Q is (1, 4). Fin…
37A circle, $C_1$ , has the equation $x^2 + y^2 + 2x - 6y = 10$ . Find the coordinates of the centre and the exact radius of…
38The diagram shows a trapezium $ABCD$ in which $AD$ is parallel to $BC$ and angle $ABC = 90^\circ$ . The coordinates of $A$ …
39已知⊙O的半径为3，OA=4，点P是线段OA的中点，则点P与⊙O的位置关系是（） A. 点P在⊙O内 B. 点P在⊙O上 C. 点P在⊙O外 D. 以上都有可能
40A circle $C_1$ , whose centre lies on the line $2x + y = 0$ , passes through the points $P(3, -1)$ and $Q(-4, -2)$ . Find t…
41若双曲线的中心为原点， $F(0,-2)$ 是双曲线的焦点，过 $F$ 的直线 $l$ 与双曲线相交于 $M$ ， $N$ 两点，且 $MN$ 的中点为 $P(3,1)$ ，则双曲线的方程为 $(\quad)$ A. $\dfrac{x^{2}}{3}-y^{2}=1$ …
42A curve has equation $y = x^2 - x - 12$ . Points A and B lie on the curve and have x-coordinates of $k$ and $2k$ respecti…
43A circle $C$ has equation $x^2 + y^2 + 6x - 4y = 12$ . Find the centre and the radius of $C$ . The points $P(-8, 2)$ and $Q(1, -1)$ …
44The equation of a curve is $y = 2x^2 - 6x + 9$ . By expressing $2x^2 - 6x + 9$ in the form $a(x+b)^2 + c$ , where $a$ , $b$ …
45The diagram shows a kite $ABCD$ in which the line $AC$ passes through the origin $O$ and has a gradient of 1.5. Find the…
46A circle, $C$ , has equation $x^2 + y^2 - 6x + 4y - 12 = 0$ . Find the radius and the coordinates of the centre of $C$ . Th…
47The highest point on a circle $C_1$ is (2, 8). The equation of the tangent, $T$ , to $C_1$ at the point (6, 6) is $3y + 4x = 42$ …
48Solutions to this question by accurate drawing will not be accepted. The diagram (not drawn to scale) shows a trapezium…
49The curve $x^2 + xy - 6y^2 + 9 = 0$ and line $y = x + 1$ intersect at the points $R$ and $S$ . Find the $x$ -coordinate of…
50The gradient function of a curve is given by $\frac{dy}{dx} = -\frac{1}{x^2} + 6 \cos 3x$ . The coordinates of the point…
51The diagram shows part of the curve $y = 3x^{2}$ . The points P and Q lie on the curve, and have the same y-coordinates.…
52已知点 $P(x_{0},y_{0})$ 是曲线 $C_{1}$ 上的动点，若抛物线 $C_{2}$ ： $y^{2}=4x$ 上存在不同的两点 $A$ 、 $B$ 满足 $PA$ 、 $PB$ 的中点均在 $C_{2}$ 上，则 $A$ …
53Sketch, on the axes below, the graphs of $y = -3\sin x$ and $y = 1 - \frac{2x}{\pi}$ for $0 \le x \le 2\pi$ . Find the nu…
54已知椭圆 $\dfrac{x^{2}}{3}+\dfrac{y^{2}}{4}=1$ 的弦被点 $(1,1)$ 平分，那么这条弦所在的直线方程为 $(\quad)$ A. $4x+3y-7=0$ B. $4x-3y-7=0$ C.…
55The diagram shows the quadrilateral $ABCD$ in which point $A$ is $(-1, 2)$ and point $C$ is $(8.5, 4)$ . The point $P(6.5, 8)$ …
56Points $P(-2, 0)$ , $Q(5, 1)$ , $R(6, -6)$ and $S(-1, -7)$ lie on a circle. $PQ$ and $SR$ are parallel chords of the same…
57The equation of a circle, $S_1$ , with centre $C$ is given by $x^2 + y^2 - 8x - 16y + 55 = 0$ . A second circle, $S_2$ , wi…
58Solutions to this equation by accurate drawing will not be accepted. $ABC$ is a triangle with vertices $A(0, h)$ , $B(2h, -2)$ …
59In the diagram, $AB$ is parallel to $DC$ and the coordinates of $A$ , $C$ and $D$ are $(-4, 1)$ , $(3, -1)$ and $(-1, -4)$ …
60Points $A(3, 8)$ and $B(6, 5)$ are on a circle. Find the equation of perpendicular bisector of $AB$ . Given that $y + x = 8$ …