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The question is from CAT Coordinate Geometry. This is about Points equidistant from lines. We need to find out the equation of a line which can be formed from the points which are equidistant from another given line. Take Geometry, add one unit of algebra; take a diagram, explain it with x's and y's. For the purists, it is geometry without the romance, for the pragmatists it is Geometry with expanded scope. CAT exam does test one on ideas from CAT Coordinate Geometry once in a while. It is important to cover ideas from Coordinate Geometry in your CAT Preparation

Question 4: What is the equation of a set of points equidistant from the lines y = 5 and x = –4?

x + y = –1
x – y = –1
x + y = 1
–x + y = –1

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Explanatory Answer

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Explanatory Answer

Method of solving this CAT Question from CAT Coordinate Geometry : Set of points equidistant from two straightlines must lie on the angle bisector of the line angle formed at the point of intersection. Try proving this. Hint: Use congruence.

Let us try to draw the given lines on the coordinate plane.
CAT : Geometry: Coordinate Geometry Line

A set of points equidistant from the given two lines should lie on the dotted line as indicated. You can think of it as the perpendicular bisector to the base of an isosceles triangle formed by (–4, 5) and the two points on x = –4 and y = 5.

Or, the set of points equidistant from two lines form the angle bisector of the angle formed at the point of intersection of the two lines. The angle between these two lines is 900. Importantly, the lines are parallel to the axes. So, thinking of the line that is the angle bisector of this angle should not be too difficult.

This dotted line is at an angle of 135^o with respect to the positive direction of x–axis and also passes through (–4, 5).

Slope = m = tan (135^o) = –1.
Therefore, the equation is given by (y – y₁) = m

(x – x₁) where (x₁, y₁) is (–4, 5).
(y – 5) = –(x + 4)
x + y = 1