# CAT Practice : 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.

## Angle bisector theorem

Q.4: What is the equation of a set of points equidistant from the lines y = 5 and x = –4?
1. x + y = –1
2. x – y = –1
3. x + y = 1
4. –x + y = –1

Choice C. x + y = 1

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## Detailed Solution

Let us try to draw the given lines on the coordinate plane.

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 135o with respect to the positive direction of x–axis and also passes through (–4, 5).

Slope = m = tan (135o) = –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

Correct Answer: x + y = 1

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## More questions from Coordinate Geometry

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.