67. Show that the equation of the line passing through the origin and making an angle ø with the line <math><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi><mi>i</mi><mi>s</mi><mfrac><mrow><mi>y</mi></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mo>±</mo><mi>t</mi><mi>a</mi><mi>n</mi><mo>,</mo></mrow><mrow><mi>m</mi><mo>?</mo><mi>t</mi><mi>a</mi><mi>n</mi><mo>,</mo></mrow></mfrac><mo>.</mo></mrow></math>

67. The given eqn of line is.l1 : y = mx + c.Slope of l1 = mLet m? be the slope of line passing through origin (0, 0) and making angle θ with l1Thus, (y 0) = m? (x 0)y = m? xm? = yx______ (1)And tanθ = |m′−m1+m′·m| = m′−m1+m′mWhen, tanθ = m′−m1+m′m.tanθ + m? m tanθ = m’ - mm + tanθ = m? - m?m tanθm' = m+tanθ1−mtanθ.When tan θ = −(m′−m1+m′m)tan θ + m? m tanθ = -m? + mm' = m−tanθ1+mtanθ.Hence combining the two we get,m′=m±tanθ1?mtanθ.⇒yx=m±tanθ1?mtanθ. {-: eqn (1) }

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

67. Show that the equation of the line passing through the origin and making an angle ø with the line $y = m x + c i s \frac{y}{x} = \frac{m \pm t a n,}{m ? t a n,} .$

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alok kumar singh | Contributor-Level 10

4 months ago

67. The given eqⁿ of line is.

l₁ : y = mx + c.

Slope of l₁ = m

Let m? be the slope of line passing through origin (0, 0) and making angle θ with l₁

Thus, (y 0) = m? (x 0)

y = m? x

m? = $y$
$x$ ______ (1)

And tanθ = $| \frac{m^{'} - m}{1 + m^{'} \cdot m} |$ = $\frac{m^{'} - m}{1 + m^{'} m}$

When, tanθ = $\frac{m^{'} - m}{1 + m^{'} m} .$

tanθ + m? m tanθ = m’ - m

m + tanθ = m? - m?m tanθ

m' = $\frac{m + t a n θ}{1 - m t a n θ} .$

When tan θ = $- (\frac{m^{'} - m}{1 + m^{'} m})$

tan θ + m? m tanθ = -m? + m

m' = $\frac{m - t a n θ}{1 + m t a n θ} .$

Hence combining the two we get,

$m^{'} = \frac{m \pm t a n θ}{1 ? m t a n θ} .$

$\Rightarrow \frac{y}{x} = \frac{m \pm t a n θ}{1 ? m t a n θ} .$ {-: eqⁿ (1) }

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67. Show that the equation of the line passing through the origin and making an angle ø with the line $y = m x + c i s \frac{y}{x} = \frac{m \pm t a n,}{m ? t a n,} .$

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67. Show that the equation of the line passing through the origin and making an angle ø with the line $y = m x + c i s \frac{y}{x} = \frac{m \pm t a n,}{m ? t a n,} .$