Let the equation x(x+2)(12-k)=2 have equal roots. Then the distance of the point (k, k/2) from the line 3 x+4 y+5=0 is

Solution

Maths sir · 4 steps

Equal roots means the discriminant is zero. That gives k, and then it is a plain distance formula.

1. Make it quadratic. Expand and move 2 across to get (12-k)x² + 2(12-k)x - 2 = 0. For equal roots this must be a genuine quadratic with discriminant zero.

   4(12-k)² + 8(12-k) = 4(12-k)(14-k) = 0

2. Pick the right k. k = 12 kills the x² term and leaves -2 = 0, which has no roots at all. So k = 14, and the point is (14, 7).
3. Distance formula. Distance = (|3(14) + 4(7) + 5|)/√(3² + 4²) = (|42 + 28 + 5|)/5 = 75/5.
4. So. The distance is 15, option (a).

The idea to keep. Equal roots means discriminant zero, but always check that the leading coefficient is not zero. A value of k that removes the x² term must be rejected.

Answer from the NTA final answer key. Solution by PrepWiser.