Straight Lines: Mixed Practice

Before you start#

Twelve ques­tions on straight lines. A small habit that saves a lot of trou­ble: sketch the points roughly before you cal­cu­late any­thing. Then look at what the ques­tion gives you and let that decide which form of the equa­tion to use.

The ques­tions#

1. Find the equa­tion of the line through $(-2, 3)$ with incli­na­tion $135^\circ$.
2. Find $p$ if $(p, 1)$, $(3, 5)$ and $(7, 13)$ are collinear.
3. Find the equa­tions of the medi­ans of the tri­an­gle $A(0, 0)$, $B(6, 0)$, $C(2, 8)$ and ver­ify that they meet at $\displaystyle (\tfrac{8}{3}, \tfrac{8}{3})$.
4. Find the point on the $x$-axis equidis­tant from $(1, 4)$ and $(5, 2)$.
5. Find the equa­tion of the line through the inter­sec­tion of $2x + y = 5$ and $x - 3y = -1$ and per­pen­dic­u­lar to $x + y = 0$.
6. Find the angle between $\sqrt{3}x - y = 2$ and $x - \sqrt{3}y = 5$.
7. Find the image of $(1, 2)$ in the line $x - 3y + 4 = 0$.
8. Find the equa­tions of the lines through $(1, 2)$ mak­ing an angle of $45^\circ$ with $3x - y = 7$.
9. Find the dis­tance of the point $(2, 3)$ from the line $2x - 3y + 9 = 0$ mea­sured along the line $x - y + 1 = 0$.
10. Show that the area of the tri­an­gle formed by $y = m_1x + c_1$, $y = m_2x + c_2$ and $x = 0$ is $\displaystyle \frac{(c_1 - c_2)^2}{2\lvert m_1 - m_2 \rvert}$.
11. A stu­den­t's monthly phone bill comes to ₹$250$ when she uses $300$ min­utes and ₹$370$ when she uses $500$ min­utes. If the bill grows lin­early with the min­utes used, what is the fixed charge?
12. Find the equa­tion of the line whose per­pen­dic­u­lar dis­tance from the ori­gin is $4$ and whose nor­mal makes $30^\circ$ with the pos­i­tive $x$-axis.

Answers to check against#