(B) The given line is $\frac{x}{a} - \frac{y}{b} - 1 = 0$. Multiplying by $ab$,we get $bx - ay - ab = 0$. The length of the perpendicular from point $

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$\left| \frac{b^2 - ab - a^2}{\sqrt{a^2 + b^2}} \right|$

(B) The given line is $\frac{x}{a} - \frac{y}{b} - 1 = 0$. Multiplying by $ab$,we get $bx - ay - ab = 0$. The length of the perpendicular from point $(x_1, y_1) = (b, a)$ to the line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. Substituting the values,$d = \frac{|b(b) - a(a) - ab|}{\sqrt{b^2 + (-a)^2}} = \frac{|b^2 - a^2 - ab|}{\sqrt{a^2 + b^2}}$. Thus,the length is $\left| \frac{b^2 - ab - a^2}{\sqrt{a^2 + b^2}} \right|$.

Class 11 Mathematics Straight Line Distance between two lines, Perpendicular distance of the line from a point, Position of point w.r.t. line

Easy

The length of the perpendicular from the point $(b, a)$ to the line $\frac{x}{a} - \frac{y}{b} = 1$ is:

A
$\left| \frac{a^2 - ab + b^2}{\sqrt{a^2 + b^2}} \right|$
B
$\left| \frac{b^2 - ab - a^2}{\sqrt{a^2 + b^2}} \right|$
C
$\left| \frac{a^2 + ab - b^2}{\sqrt{a^2 + b^2}} \right|$
D
None of these

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Straight Line

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Distance between two lines, Perpendicular distance of the line from a point, Position of point w.r.t. line

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DifficultJEE MAIN 2025

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DifficultTS EAMCET 2013

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Medium

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The length of the perpendicular from the point $(b, a)$ to the line $\frac{x}{a} - \frac{y}{b} = 1$ is:

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