A fixed point P(10, 0) lies on the circle x^2 + y^2 = 100 . The number of integral values of c for which the line 3x + y = c bisects exactly two distinct chords drawn from P to the circle, is :

Step-by-step solution

The locus of the midpoints of all chords drawn from the point P(10, 0) to the circle x^2 + y^2 = 100 is a circle. Let the midpoint be (h, k) . The equation of the chord with midpoint (h, k) is given by T = S₁ : hx + ky - 100 = h^2 + k^2 - 100 hx + ky = h^2 + k^2 Since the chord passes through P(10, 0) , we substitute x = 10, y = 0 : 10h = h^2 + k^2 h^2 - 10h + k^2 = 0 This represents a circle with center (5, 0) and radius 5 . For the line 3x + y - c = 0 to bisect two distinct chords, it must intersect this locus ci