Coordinate geometry › Straight lines

Angle of inclination

Solve the first problem 17 problems

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1. Parameter for a vertical line ●○○○○ 10 ptsGiven points $A(1+a,\,2a)$ and $B(1-a,\,3)$, the angle of inclination of line $AB$ is $90^\circ$. Find $a$.
2. Angle between two lines on a surface ●○○○○ 10 ptsOn a surface with first fundamental form $ds^2=du^2+dv^2$, find the angle (in degrees) between the curves $v=u$ and $v=-u$ at their point of intersection.
3. Inclination angle of a vertical line ●○○○○ 10 ptsThe line $l$ is given by $x = \tan\left(-\frac{\pi}{6}\right)$. Find the inclination angle of $l$ in degrees.
4. Angle between y = x² and y = 1/x ●●○○○ 20 ptsFind the acute angle of intersection, in degrees, between the curves $y = x^2$ and $y = \dfrac{1}{x}$, measured between their tangents at the point of intersection. Round to two decimal places.
5. Angle with z-axis from two axis angles ●●○○○ 20 ptsA line makes an angle of $60^\circ$ with each of the $x$-axis and the $y$-axis. Find the acute angle, in degrees, that the line makes with the $z$-axis.
6. Angle between two lines ●●○○○ 20 ptsFind the angle between the lines $l_1: \sqrt{3}x - y + 2 = 0$ and $l_2: 3x + \sqrt{3}y - 5 = 0$, in degrees.
7. Counting lines with acute inclination ●●●○○ 50 ptsLet the line $ax+by+c=0$ have $a$, $b$, $c$ as three different elements of the set $\{-3,-2,-1,0,1,2,3\}$, and suppose the inclination angle of the line is acute. Find the number of distinct such lines.
8. Rotating a line to equal intercepts ●●●○○ 50 ptsThe line $y=10-2x$ is rotated anticlockwise about the origin through an angle $\theta$, where $0 \le \theta \le 90$ (in degrees). The image has $x$-intercept $(p,0)$ and $y$-intercept $(0,q)$. Determine $\theta$ such that $p=q$. Round to two decimal places.
9. Coefficient from line-plane angle ●●●○○ 50 ptsThe angle between the plane $3x + by - 5z = 2$ and the straight line $x + 2 = \frac{3 - y}{2} = \frac{z - 1}{2}$ is $45^\circ$. Find the constant $b$, given that $0 < b < 10$.

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