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The Straight Lines — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsThe Straight Lines2 Marks
Question
A straight line drawn through the point A (2, 1) making an angle $\frac{\pi}{4}$ with positive x-axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.
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Answer

The equation of line is$\frac{\text{x}-2}{\cos\alpha}=\frac{\text{y}-1}{\sin\alpha}=\text{r}$
$\Rightarrow\frac{\text{x}-2}{\frac{1}{\sqrt2}}=\frac{\text{y}-1}{\frac{1}{\sqrt2}}=\text{r}$
or $\text{x}=\frac{1}{\sqrt2}\text{r}+2,\ \text{y}=\frac{1}{\sqrt2}\text{r}+1$
$\text{B}\Big(\frac{\text{r}}{\sqrt2}+2,\ \frac{\text{r}}{\sqrt2}+1\Big)$ lie on x + 2y + 1 = 0
$\therefore\frac{\text{r}}{\sqrt2}+2+\frac{2\text{r}}{\sqrt2}+2+1=0$
$\frac{3\text{r}}{\sqrt2}=\pm5$
$\text{r}=\frac{5\sqrt2}{3}$
The lenght of AB is $\frac{5\sqrt2}{3}$ units.

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