# CAT Practice : Coordinate Geometry

Open-ended questions that ask us to find points that are closest to a line or farthest from an arc are the toughest. It is important to find the best starting point in these kinds of questions.

## Straightline Shortest distance

Q.1: Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?
1. 2
2. 3
3. 5
4. 4

Choice D. 4

## Detailed Solution

Any point on the line ${{x \over 3} + {y \over 5}}$ = 1 will have the shortest overall distance. However, we need to have integral coordinates. So, we need to find points with integral coordinates as close as possible to the line 5x + 3y = 15.

Substitute x =1, we get y = 2 or 3
Substitute x = 2, we get y = 1 or 2

Sum of distances for (1, 2) = ${\sqrt {8}} + {\sqrt {10}}$
Sum of distances for (1, 3) = ${\sqrt {13}} + {\sqrt {5}}$
Sum of distances for (2, 1) = ${\sqrt {2}} + {\sqrt {20}}$
Sum of distances for (2, 2) = ${\sqrt {5}} + {\sqrt {13}}$

${\sqrt {5}} + {\sqrt {13}}$ is the shortest distance.
Sum of abscissa + ordinate = 4

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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.