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Line whose perpendicular from the origin makes a given angle and which cuts off a triangle of given area with the axes

The same question, asked 2 times: 30 Jan 2023*, 2 Apr 2025. Every wording below, oldest first.

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  1. Question 1

    30 Jan 2023

    61
    A straight line cuts off the intercepts OA=aOA = a and OB=bOB = b on the positive directions of xx-axis and yy axis respectively. If the perpendicular from origin OO to this line makes an angle of π6\frac{\pi}{6} with positive direction of yy-axis and the area of △OAB\bigtriangleup OAB is 9833\frac{98}{3}\sqrt{3}, then a2−b2a^{2}-b^{2} is equal to:

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  2. Question 2

    2 Apr 2025

    144
    Let the area of the triangle formed by a straight Line L:x+by+c=0L:x+by+c= 0 with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line LL makes an angle of 45∘45^{\circ} with the positive xx-axis, then the value of b2+c2b^{2}+c^{2} is:

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