3.7 Parametric form

  Straight Lines

• Parametric equations of straight line $AB$ is $$\begin{equation} \begin{aligned} x = {x_1} + r\cos \theta \\ y = {y_1} + r\sin \theta \\\end{aligned} \end{equation} $$
• If point $P$ lies above point $Q$, then $r$ is positive and coordinates of $P$ are $$({x_1} + r\cos \theta ,{y_1} + r\sin \theta )$$ and if point $P$ lies below point $Q$, then $r$ is negative and coordinates of $P$ are$$({x_1} - r\cos \theta ,{y_1} - r\sin \theta )$$

Question 3. Find the equation of line through the point $A(2,3)$ and making an angle of ${45^ \circ }$ with the $X-$axis. Also determine the length of intercept on it between $A$ and the line $x+y+1=0$.

Solution: The equation of line through $A$ and making an angle of ${45^ \circ }$ with the $X-$axis is $$\begin{equation} \begin{aligned} \frac{{x - 2}}{{\cos {{45}^ \circ }}} = \frac{{y - 3}}{{\sin {{45}^ \circ }}} \\ x - y + 1 = 0 \\\end{aligned} \end{equation} $$

Suppose this line meets the line $x+y+1=0$ at $P$ such that $AP=r$. Then the co-ordinates of $P$ are given by $$\begin{equation} \begin{aligned} \frac{{x - 2}}{{\cos {{45}^ \circ }}} = \frac{{y - 3}}{{\sin {{45}^ \circ }}} = r \\ x = 2 + r\cos {45^ \circ },{\text{ }}y = 3 + r\sin {45^ \circ } \\ x = 2 + \frac{r}{{\sqrt 2 }},{\text{ }}y = 3 + \frac{r}{{\sqrt 2 }} \\\end{aligned} \end{equation} $$

Thus, the co-ordinates of $P$ are $$(2 + \frac{r}{{\sqrt 2 }},3 + \frac{r}{{\sqrt 2 }})$$

Since $P$ lies on $x+y+1=0$, so $$\begin{equation} \begin{aligned} 2 + \frac{r}{{\sqrt 2 }} + 3 + \frac{r}{{\sqrt 2 }} + 1 = 0 \\ r = - 3\sqrt 2 \\\end{aligned} \end{equation} $$

Therefore, length of intercept $AP = \left| r \right| = 3\sqrt 2 $.