11. The slope of a line is double of the slope of another line. If tangent of the angle between them is <math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math> , find the slopes of the lines.

11. Let the two slopes be m and 2m. And if θ is the angle between the two lines.tanθ=|m1−m21+m1m2|⇒13=|2m−m1+2m..m|=|m1+2m2|⇒m1+2m2=±13 (|x|=±x)Case I. When m1+2m=13⇒3m=2m2+1⇒2m2−3m+1=0⇒2m2−2m−m+1=0⇒2m(m−1)−(m−1)=0⇒(m−1)(2m−1)−0⇒m=1 m=12Case II. When m1+2m2=−13⇒3m=−(1+2m2)⇒3m=−1−2m2⇒2m2+3m+1=0⇒2m2+2m+m+1=0⇒2m(m+1)+(m+1)=0⇒(m+1)(2m+1)=0⇒m=−1 m=−12Hence, m=1,12,−1 m=−12∴ The possible slopes of the two lines (m, 2m) are (1, 2), (12,1),(−1,−2) and (−12,−1)

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Ncert Solutions Maths class 11th Maths Class 11th Straight Lines

11. The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$ , find the slopes of the lines.

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alok kumar singh | Contributor-Level 10

4 months ago

11.

Let the two slopes be m and 2m. And if θ is the angle between the two lines.

$t a nθ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$

$\Rightarrow \frac{1}{3} = | \frac{2 m - m}{1 + 2 m . . m} | = | \frac{m}{1 + 2 m^{2}} |$

$\Rightarrow \frac{m}{1 + 2 m^{2}} = \pm \frac{1}{3} ($ $| x | = \pm x)$

Case I. When $\frac{m}{1 + 2 m} = \frac{1}{3}$

$\Rightarrow 3 m = 2 m^{2} + 1$

$\Rightarrow 2 m^{2} - 3 m + 1 = 0$

$\Rightarrow 2 m^{2} - 2 m - m + 1 = 0$

$\Rightarrow 2 m (m - 1) - (m - 1) = 0$

$\Rightarrow (m - 1) (2 m - 1) - 0$

$\Rightarrow m = 1 m = \frac{1}{2}$

Case II. When $\frac{m}{1 + 2 m^{2}} = - \frac{1}{3}$

$\Rightarrow 3 m = - (1 + 2 m^{2})$

$\Rightarrow 3 m = - 1 - 2 m^{2}$

$\Rightarrow 2 m^{2} + 3 m + 1 = 0$

$\Rightarrow 2 m^{2} + 2 m + m + 1 = 0$

$\Rightarrow 2 m (m + 1) + (m + 1) = 0$

$\Rightarrow (m + 1) (2 m + 1) = 0$

$\Rightarrow m = - 1 m = - \frac{1}{2}$

Hence, $m = 1, \frac{1}{2}, - 1 m = - \frac{1}{2}$

$∴$ The possible slopes of the two lines (m, 2m) are (1, 2), $(\frac{1}{2}, 1), (- 1, - 2)$ and $(- \frac{1}{2}, - 1)$

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11. The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$ , find the slopes of the lines.

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11. The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$ , find the slopes of the lines.