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

Our Online Course, Now on Google Playstore!

Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.

Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

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.