Show that the tangent of an angle between the lines x - a - y - b -1 and x - a - y - b -1 is 2 ab - a 2- b 2

Let the line x/a+y/b=1 be AB and the line x/a y/b = 1 be CD. Equation of AB, bx+ay/ab=1 ⇒ ay = bx+ab ⇒ y = bx/a+b Therefore, m_1 = b/a Similarly, the equatio ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Angle between Two Lines

Show that the...

Question

Show that the tangent of an angle between the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{a} - \frac{y}{b} = 1$ is $\frac{2 a b}{a^{2} - b^{2}}$

Open in App

Solution

Let the line $\frac{x}{a} + \frac{y}{b} = 1$ be AB and the line $\frac{x}{a} - \frac{y}{b} = 1$ be CD.

Equation of AB, $\frac{b x + a y}{a b} = 1$

$\Rightarrow a y = - b x + a b$

$\Rightarrow y = - \frac{b x}{a} + b$

Therefore, $m_{1} = - \frac{b}{a}$

Similarly, the equation of CD, $\frac{b x - a y}{a b} = 1$

$\Rightarrow b x - a y = a b$

$\Rightarrow a y = \frac{b x}{a} - a$

Therefore, $m_{2} = \frac{b}{a}$

The tangent of angle between the lines AB and CD is

$t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣ = ∣ ∣ ∣ ∣ \frac{- \frac{b}{a} - \frac{b}{a}}{1 + (- \frac{b}{a}) (\frac{b}{a})} ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ \frac{\frac{- 2 b}{9}}{\frac{a^{2} - b^{2}}{a}} ∣ ∣ ∣ = ∣ ∣ \frac{- 2 a b}{a^{2} - b^{2}} ∣ ∣$

The tangent of the angle between the lines $= \frac{2 a b}{a^{2} - b^{2}}$

Suggest Corrections

Similar questions

Q. Show that the tangent of an angle between the lines

\frac{x}{a} + \frac{y}{b} = 1 a n d \frac{x}{a} - \frac{y}{b} = 1 i s \frac{2 a b}{a^{2} - b^{2}}

Q. Show that the tangent of an angle between the lines

\frac{x}{a} + \frac{y}{b} = 1 and \frac{x}{a} - \frac{y}{b} = 1 is \frac{2 a b}{a^{2} - b^{2}}

Q. Show that

\frac{x^{2} + 2 x y + y^{2} - a^{2} + 2 a b - b^{2}}{(x + y - a + b)} = (x + y + a - b)

Q. Prove that the lines:

(a + b) x + (a - b) y - 2 a b = 0....... (1)

(a - b) x + (a + b) y - 2 a b = 0....... (2)

and

x + y = 0...... (3)

form an isosceles triangle whose vertical angle is

2 {tan}^{- 1} (a / b)

If A = $[\begin{matrix} a & b b & a \end{matrix}]$ and $A^{2}$ = $[\begin{matrix} x & y y & x \end{matrix}]$ , then

Join BYJU'S Learning Program

Show that the tangent of an angle between the lines xa+yb=1 and xa−yb=1 is 2aba2−b2

Show that the tangent of an angle between the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{a} - \frac{y}{b} = 1$ is $\frac{2 a b}{a^{2} - b^{2}}$