GMAT Questions | Coordinate Geometry

You will get two to three questions from Coordinate Geometry in the GMAT quant section. Questions appear in both variants viz., problem solving and data sufficiency. The concepts tested include finding the slope, identifying the quadrants through which a line passes, finding the length of a line segment, determining the equation of a line, equations of parallel and perpendicular lines, area of triangle, and equation of circles. Sample GMAT practice questions in coordinate geometry is given below. Attempt these questions and check whether you have got the correct answer. If you have not, go to the explanatory answer or the video explanations to learn how to crack the question.

Curated sample GMAT questions from these topics are presented here below. Attempt these questions and check if you have got the correct answer. One can use the explanatory answer or Video explanation made available (wherever provided) to help crack important GMAT quant questions.

1. What is the equation of a circle of radius 6 units centered at (3, 2)?

   1. x² + y² + 6x - 4y = 23
   2. x² + y² - 6x + 4y = 23
   3. x² + y² + 6x + 4y = 23
   4. x² + y² - 6x - 4y = -23
   5. x² + y² - 6x - 4y = 23

   Correct AnswerChoice E
   The equation of the circle is x² + y² - 6x - 4y = 23

   Explanation Video Solution
   Hint to solve this sub 600 level GMAT question

   Step 1: Equation of a circle whose center is at (a, b) and whose radius is 'r' units is (x - a)² + (y - b)² = r²
   Step 2: In this question, a = 3, b = 2, and r = 6. Substitute and derive the equation of the circle.

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2. 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 (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?

   1. 1
   2. 5
   3. 11
   4. 8
   5. 3

   Correct AnswerChoice E
   3 such points exist in set S.

   Explanation Video Solution
   Hint to solve this 700 level GMAT Problem

   Step 1: Key Point: The sum of the distances is minimum only when point P lies on the line segment joining the intercepts.
   Step 2: Find the equation of the line with (8, 0) and (0, 12) as x and y intercepts respectively.
   Step 3: Substitute positive integer values for x from 0 to 8 in the equation obtained in step 2 and check whether corresponding y values are integers.
   Step 4: Count number of values for which both x and y coordinates are integers.

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3. Line L is perpendicular to line K whose equation is 3y = 4x + 12; Lines L and K intersect at (p, q).
   Is p + q > 0?

   1. x intercept of Line L is less than that of Line K
   2. y intercept of Line L is less than that of Line K

   Correct AnswerChoice A
   

   Explanation Video Solution
   Hint to solve this GMAT 700 level Coordinate Geometry DS question

   Concept: Line K is a positive sloping line. Line L, therefore, is a negative sloping line.
   Step 1: Compute x-intercept of line K and draw line L to satisfy information given in statement 1. Deduce possible points of intersection and determine whether we get a unique answer to the question.
   Step 2: Compute y-intercept of line K and draw line L to satisfy information given in statement 2. Deduce possible points of intersection and determine whether we get a unique answer to the question.
   Step 3: If we do not get a conclusive answer using either statement independently, combine the information in the two statements and determine sufficiency.

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4. Does the line x + y = 6 intersect or touch the circle C with radius 5 units?

   1. Center of the circle lies in the third quadrant.
   2. Point (-4, -4) does not lie inside the circle.

   Correct AnswerChoice E
   

   Explanation Video Solution
   Approach to solve this GMAT 700 to 750 level DS question

   Step 1 - Question Stem: The line is a negative sloping line with intercepts (6, 0), (0, 6). It passes through II, IV, and I quadrants.
   Step 2: Evaluate statement 1 by looking for a counter example. If the shortest distance between the center of the circle and the line is less than or equal to the radius, the line will touch or intersect with the circle. If a counter example exists, statement 1 is not sufficient. Else it is sufficient.
   Step 3: Look for a counter example with two different points that could be centers of the circle satisfying statement 2. If we are able to find a counter example, statement 2 will not be sufficient.
   Step 4: If we do not get a conclusive answer using either statement independently, combine the information in the two statements and determine sufficiency.

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5. Is the slope of the line that passes through the point (p, q) positive?

   1. p, q > 0
   2. The x-intercept of the line is k, such that k > p.

   Correct AnswerChoice C
   

   Explanation Video Solution
   Approach to solve this GMAT 650 to 700 Coordinate Geometry DS question

   Step 1: Evaluate Statement 1 Alone: We know point (p, q) is a I quadrant point. Can we deduce the slope of the line with this information?
   Step 2: Evaluate Statement 2 Alone: The x-intercept of line k is greater the x-coordinate of the point. Try to describe two lines one with positive slope and the other with a negative slope satisfying this condition. If it is possible statement 2 alone is not sufficient.
   Step 3: If we do not get a conclusive answer using either statement independently, combine the information in the two statements and determine sufficiency.

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