Math topic中文

Parallel perpendicular

condition for two lines to be parallel or perpendicular

Practice questions

1A curve has the equation $y = \frac{2x - 3}{3x + 4}$ , where $x \ne -\frac{4}{3}$ . The normal to the point $P$ on the cur…
2The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
3The point $P$ lies on the curve $y = \ln\left(\frac{x+1}{x-1}\right)$ for $x > 1$ . The normal to the curve at $P$ is par…
4The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
5The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
6已知函数 $f(x)=x^{2}-ax(a\neq 0)$ ， $g(x)=\ln x$ ， $f(x)$ 图象与 $x$ 轴异于原点的交点 $M$ 处的切线为 $l_{1}$ ， $g(x-1)$ 与 $x$ 轴的交点 $N$ 处的切线为…
7The equation of a circle is $x^2 + y^2 + 8x - 20y - 53 = 0$ . Find the radius of the circle and the coordinates of its ce…
8A circle with centre O passes through the points P(-1,7) and Q(0,8). State the relationship between the perpendicular bi…
9Two 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…
10The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
11The equation of a circle $C_1$ is given by the equation $x^2 + y^2 - 4x - 8y = 0$ . Find the coordinates of the center an…
12若直线 $\begin{cases}{x=1-t}\\ {y=2+3t}\end{cases}(t$ 为参数 $)$ 与直线 $y=kx-1$ 平行，则常数 $k=(\quad)$ A. $-3$ B. $-\dfrac{1}{3}$ …
13A(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…
14The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
15Given the curve $y = \frac{1}{2}e^{2x} - \frac{1}{3}e^{-3x} + \frac{1}{3}$ and the normal to the curve at point $P\left(\ln 2, \frac{55}{24}\right)$ …
16The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
17已知圆 $C$ ： $(x+1)^{2}+(y-1)^{2}=1$ 与 $x$ 轴切于 $A$ 点，与 $y$ 轴切于 $B$ 点，设劣弧 $\hat AB$ 的中点为 $M$ ，则过点 $M$ 的圆 $C$ 的切线方程是 $()$ A…
18The diagram shows part of the curve $y = \frac{1}{\left(\frac{3}{4}x - 5\right)^2} + 1$ . $P$ is the point on the curve w…
19Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
20The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
21Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows…
22Solutions to this question by accurate drawing will not be accepted. The diagram shows a trapezium $ABCD$ in which $AB$ …
23Solutions to this question by accurate drawing will not be accepted. The diagram shows an isosceles triangle PQR in whic…
24In the diagram, $OPQR$ is a trapezium with $OP$ parallel to $RQ$ and $OP$ perpendicular to $PQ$ . The point $P$ has coord…
25The points $P$ and $Q$ both lie on a circle and have coordinates $(-10, 15)$ and $(10, 0)$ respectively. The centre of t…
26The line $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ and $b$ are positive constants, intersects the x-axis at A and the y…
27点 $(a,b)$ 关于直线 $x+y+1=0$ 的对称点是___.
28已知三个点 $A(0,0)$ ， $B(2,0)$ ， $C(4,2)$ ，则 $\triangle ABC$ 的外心的纵坐标是 $(\quad)$ A. $\dfrac{3}{2}$ B. $3$ C. $\dfrac{5}{2}$ …
29The line $y + 2x = 12$ intersects the curve $y = 2x^2 - 6x - 4$ at two distinct points $A$ and $B$ . Find the coordinates…
30The diagram shows part of the curve $y = \sin x$ . A is a point on the curve that lies on the line $x = \frac{\pi}{3}$ . T…
31The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
32 $ABCD$ is a parallelogram. Point $A$ lies on the $y$ -axis and coordinates of $D$ is $(6, -1)$ . $AC$ is a horizontal line…
33The gradient function of a curve is given by $\frac{\text{d}y}{\text{d}x} = \frac{16}{x^3} - 4$ . Given that the line $2y - x - 6 = 0$ …
34The equation of a curve is $y = ax^2 + 4x + 3a - 7$ , where $a$ is a constant. The line $y = 2ax - 5$ is a tangent to the…
35The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
36A circle, with centre $C$ , has equation $x^2 + y^2 -10x -4y +25 = 0$ . Find the coordinates of $C$ and the radius of the…
37The graph shows part of the curve $y = -3(2x-4)^{-1/3}$ . The point $P(-2,\ 1.5)$ lies on the curve and the normal to the…
38The diagram shows part of the curve $y = \sqrt{2x + 4}$ , meeting the $x$ -axis at the point $A$ and the $y$ -axis at the p…
39The figure shows three points $A$ , $C$ and $D$ whose coordinates are $(3, 5)$ , $(4, -2)$ and $(7, 2)$ respectively. Find…
40The diagram (not drawn to scale) shows a right-angled triangle PQR in which point P is (5, 4) and point R is (9, 2). Poi…
41Solutions to this question by accurate drawing will NOT be accepted. The figure shows a right-angled triangle $ABC$ ,…
42若曲线 $y=e^{-x}$ 上点 $P$ 处的切线平行于直线 $2x+y+1=0$ ，则点 $P$ 的坐标为 ___ ．
43The diagram shows a quadrilateral $ABCD$ with vertices $A(-4, -3)$ , $B(6, 1)$ , $C(3, 5)$ and $D(-2, 3)$ . $M$ which lies…
44Point $P$ lies on the curve $y = \frac{kx^2}{\sqrt{2-x}}$ , where $k$ is a constant. The $x$ -coordinate of $P$ is $-2$ . G…
45The diagram shows part of the curve $y = x^3 + 3x^2 - 2$ . The straight line $L$ cuts the curve at $A(-3, -2)$ , the x-axi…
46The diagram shows a trapezium with vertices A(6, 8), B, C and D. The sides AB and DC are parallel and angle ABC is $90^\circ$ …
47The diagram shows points $A(5, 0)$ , $B(a, 3a)$ and $C(-2, 6)$ such that the line $AB$ is perpendicular to the line $BC$ .…
48求出满足下列条件的直线方程. (1)经过点 $A(-3,2)$ 且与直线 $x+3y-4=0$ 垂直； (2)经过点 $B(2,7)$ 且与直线 $2x-y-4=0$ 平行.
49Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
50The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
51The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
52In the diagram below, the line $OP$ makes an angle of $\alpha^\circ$ with the positive $x$ -axis such that $\tan \alpha = 0.2$ …
53Solutions to this question by accurate drawing will not be accepted. $OABC$ is a trapezium with right angle $OAB$ . T…
54The points $A(5, 7)$ and $B(-11, 15)$ lie on a circle $C$ . The centre of the circle $O$ lies on the line $y = -2x - 3$ .…
55设曲线 $f(x)=ax^{2}$ 在点 $(2,4a)$ 处的切线与直线 $4x-y+4=0$ 垂直，则 $a=(\quad)$ A. $2$ B. $-\dfrac{1}{16}$ C. $\dfrac{1}{2}$ D.…
56Which two lines in the square grid below are parallel to each other?
57已知曲线 $y=\dfrac{|x|}{{{e}^{x}}}$ 在 $x=-1$ 处的切线和它在 $x={{x}_{0}}\left({x}_{0} > 0\right)$ 处的切线互相垂直，设 ${{x}_{0}}\in (\dfrac{m}{4},\dfrac{m+1}{4}),m\in Z$ …
58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 ___ ．
59A and B are the points of intersection of the line $4y = 2x + 1$ and the curve $3y - x = 4xy$ . Find the coordinates of A…
60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 $e$ …