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Maths-

General

Easy

Question

The angle between ax + by + c = 0 and (a + b) x – (a – b) y = 0 is

30°
45°
60°
90°

hint Hint:

First, we will find the slopes of the lines. The slope of a line $a x plus b y plus c space equals 0$ is given by $m space equals space fraction numerator negative space c o e f f i c i e n t space o f space y space over denominator c o e f f i c i e n t space o f space x end fraction$ . After that we will use the formula of the angle between the two lines $tan open parentheses theta close parentheses equals open vertical bar fraction numerator m subscript 1 minus m subscript 2 over denominator 1 plus m subscript 1 m subscript 2 end fraction close vertical bar$ where $m subscript 1$ and $m subscript 2$ are the slope of the lines and $theta$ is the angle between the lines.

The correct answer is: 45°

The formula of the angle between the two lines is given by
$tan open parentheses theta close parentheses equals open vertical bar fraction numerator m subscript 1 minus m subscript 2 over denominator 1 plus m subscript 1 m subscript 2 end fraction close vertical bar$ where $m subscript 1$ and $m subscript 2$ are the slope of the lines and $theta$ is the angle between the lines.
Step1: Finding the Slopes of the lines.
The slope of a line $a x plus b y plus c space equals 0$ is given by $m space equals space fraction numerator negative space c o e f f i c i e n t space o f space y space over denominator c o e f f i c i e n t space o f space x end fraction$ hence, $m equals fraction numerator negative b over denominator a end fraction$ .
Therefore slopes of the given lines are $m subscript 1 equals fraction numerator negative b over denominator a end fraction$ and $m subscript 2 equals fraction numerator negative left parenthesis negative left parenthesis a minus b right parenthesis right parenthesis over denominator left parenthesis a plus b right parenthesis end fraction$
Step2: Finding the angles between the lines.
$tan open parentheses theta close parentheses equals open vertical bar fraction numerator m subscript 1 minus m subscript 2 over denominator 1 plus m subscript 1 m subscript 2 end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style fraction numerator negative b over denominator a end fraction end style minus begin display style fraction numerator left parenthesis a minus b right parenthesis over denominator a plus b end fraction end style over denominator 1 plus left parenthesis begin display style fraction numerator negative b over denominator a end fraction end style cross times begin display style fraction numerator left parenthesis a minus b right parenthesis over denominator left parenthesis a plus b right parenthesis end fraction end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style fraction numerator negative b cross times left parenthesis a plus b right parenthesis minus a cross times left parenthesis a minus b right parenthesis over denominator a left parenthesis a plus b right parenthesis end fraction end style over denominator begin display style fraction numerator a cross times left parenthesis a plus b right parenthesis minus b cross times left parenthesis a minus b right parenthesis over denominator a left parenthesis a plus b right parenthesis end fraction end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style negative b cross times left parenthesis a plus b right parenthesis minus a cross times left parenthesis a minus b right parenthesis end style over denominator begin display style a cross times left parenthesis a plus b right parenthesis minus b cross times left parenthesis a minus b right parenthesis end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style left parenthesis negative a b minus b squared minus a squared plus a b right parenthesis end style over denominator begin display style left parenthesis a squared plus a b minus a b plus b squared right parenthesis end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style left parenthesis negative b squared minus a squared right parenthesis end style over denominator begin display style left parenthesis a squared plus b squared right parenthesis end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar fraction numerator begin display style negative left parenthesis a squared plus b squared right parenthesis end style over denominator begin display style left parenthesis a squared plus b squared right parenthesis end style end fraction close vertical bar$
=> $tan open parentheses theta close parentheses equals open vertical bar negative 1 close vertical bar space equals greater than 1$
So $theta equals 45 to the power of ring operator$

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