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)
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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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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