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Know the shortest distance fast: from point (x1,y1) to line ax+by+c=0 use |ax1+by1+c|/√(a²+b²). Between parallel lines ax+by+c1=0 and ax+by+c2=0 use |c1−c2|/√(a²+b²). Keep signs inside absolute value, confirm lines are parallel, and plug cleanly – no extra algebra.
One of the useful tools in coordinate geometry is the ability to calculate the distance of a point from a line, and the perpendicular distance between two parallel lines. These ideas are not only simple in concept but also immensely useful in solving advanced GMAT problems. The formula for the length of a perpendicular from a point to a line gives us the shortest distance, which is often the key to unlocking geometry-based reasoning questions. Similarly, knowing how to calculate the perpendicular distance between two parallel lines helps you analyze spatial relationships with confidence. Both concepts are precise, logical, and widely tested in aptitude-based exams. Understanding these relationships deeply ensures that you approach such questions without confusion or hesitation. For a stronger foundation, it is highly recommended to opt for a high quality GMAT prep course and strengthen the basics of coordinate geometry.
The perpendicular distance is a direct and elegant way of measuring how far a point lies from a line. If the line is given in the form ax + by + c = 0 and the point is (x₁, y₁), the distance is calculated as:
The modulus ensures that the distance, being a physical measure, is never negative.
Consider the line 3x + 4y – 50 = 0 and the point (2, 5). Substituting into the formula, we get:
|3(2) + 4(5) – 50| / √(3² + 4²)
= |–24| / 5 = 4.8
Hence, the shortest distance from the point to the line is 4.8 units.
Distance Between Two Parallel Lines
When two lines are parallel, their coefficients of x and y remain identical. The perpendicular distance between them is given by:
Take the lines 3x + 4y – 50 = 0 and 3x + 4y – 10 = 0. Since the coefficients of x and y are the same, they are parallel. Applying the formula, we get:
|–50 – (–10)| / √(3² + 4²)
= |–40| / 5
= 8
Thus, the perpendicular distance between these two lines is 8 units.
Perpendicular distance teaches the discipline of taking the shortest truthful route. In GMAT preparation, measure where you stand, choose a line of effort, and verify progress with GMAT mock tests that reveal gaps precisely. Aim for fewer detours and more intentional steps. In the B-school journey, define a clear center, align your story to it, and let recommendations meet it at right angles to noise. Choose actions that can be defended with evidence and timing. For guidance that keeps choices aligned with goals, consider MBA admission consulting to shape direction. Life rewards clarity of path and steadiness of stride.
