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A line makes equal intercepts of length 'a' on the coordinate axes, intersecting the Y axis and X axis at A and B respectively. A circle is circumscribed about the triangle OAB, where O is the origin of the coordinate system. A tangent is drawn to this circle at the point O, The sum of the perpendicular distances of the vertices A, B and O from this tangent is:
  • a)
    2a
  • b)
    a/√2
  • c)
    a/2
  • d)
    a√2
  • e)
    None of these
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
A line makes equal intercepts of length 'a' on the coordinate axes, i...
Concept:
In a right-angled triangle, if the two legs are equal, then the length of the hypotenuse is equal to a√2, where a is the length of the legs.

Solution:
Let the coordinates of A and B be (0, a) and (a, 0) respectively.

Step 1: Finding the coordinates of the circumcenter O.
The midpoints of the sides OA, OB and AB are (0, a/2), (a/2, 0) and (a/2, a/2) respectively. The perpendicular bisectors of these sides pass through the circumcenter O.
The slope of the perpendicular bisector of OA is -2/a and passes through the midpoint (0, a/2). The equation of the perpendicular bisector is y = -2x/a + a/2.
The slope of the perpendicular bisector of OB is -a/2 and passes through the midpoint (a/2, 0). The equation of the perpendicular bisector is y = -ax/2 + a/2.
Solving these two equations, we get the coordinates of the circumcenter O as (a/2, a/2).

Step 2: Finding the radius of the circumcircle.
The distance between O and A is a/2 and the distance between O and B is a/2. Hence the radius of the circumcircle is a/2√2.

Step 3: Finding the equation of the tangent at O.
The equation of the circle with center O and radius a/2√2 is (x - a/2)² + (y - a/2)² = a²/8.
Differentiating this equation with respect to x, we get the slope of the tangent at O as -x/y.
At O, x = y, hence the slope of the tangent at O is -1.

Step 4: Finding the perpendicular distances of A, B and O from the tangent at O.
The equation of the tangent at O is y = -x.
The perpendicular distance of A from this tangent is |(0) - (-a)|/√2 = a/√2.
The perpendicular distance of B from this tangent is |(a) - (0)|/√2 = a/√2.
The perpendicular distance of O from this tangent is |(0) - (0)|/√2 = 0.
The sum of the perpendicular distances of A, B and O from this tangent is a/√2 + a/√2 + 0 = a√2.

Hence, the correct option is (D) a√2.
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Community Answer
A line makes equal intercepts of length 'a' on the coordinate axes, i...
Since OAB is a right angled triangle, diameter of the circle = √2a
Required sum = AX + OO + BY = diameter of circle = √2a
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A line makes equal intercepts of length 'a' on the coordinate axes, intersecting the Y axis and X axis at A and B respectively. A circle is circumscribed about the triangle OAB, where O is the origin of the coordinate system. A tangent is drawn to this circle at the point O, The sum of the perpendicular distances of the vertices A, B and O from this tangent is:a)2ab)a/√2c)a/2d)a√2e)None of theseCorrect answer is option 'D'. Can you explain this answer?
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A line makes equal intercepts of length 'a' on the coordinate axes, intersecting the Y axis and X axis at A and B respectively. A circle is circumscribed about the triangle OAB, where O is the origin of the coordinate system. A tangent is drawn to this circle at the point O, The sum of the perpendicular distances of the vertices A, B and O from this tangent is:a)2ab)a/√2c)a/2d)a√2e)None of theseCorrect answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A line makes equal intercepts of length 'a' on the coordinate axes, intersecting the Y axis and X axis at A and B respectively. A circle is circumscribed about the triangle OAB, where O is the origin of the coordinate system. A tangent is drawn to this circle at the point O, The sum of the perpendicular distances of the vertices A, B and O from this tangent is:a)2ab)a/√2c)a/2d)a√2e)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line makes equal intercepts of length 'a' on the coordinate axes, intersecting the Y axis and X axis at A and B respectively. A circle is circumscribed about the triangle OAB, where O is the origin of the coordinate system. A tangent is drawn to this circle at the point O, The sum of the perpendicular distances of the vertices A, B and O from this tangent is:a)2ab)a/√2c)a/2d)a√2e)None of theseCorrect answer is option 'D'. Can you explain this answer?.
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