Let the point P of the focal chord PQ of the parabola y²=16 x be (1,-4). If the focus of the parabola divides the chord P Q in the ratio m: n, gcd(m, n)=1, then m²+n² is equal to :

Solution

Maths sir · 4 steps

Find the parameter of P, flip it to -1/t to get Q, then measure each piece with focal distance a + x.

1. Read off the parabola. y² = 16x is y² = 4ax with a = 4, so the focus is S(4, 0). A general point is (at², 2at) = (4t², 8t).
2. Locate P and Q. For P(1, -4), 8t₁ = -4 gives t₁ = -1/2. The ends of a focal chord satisfy t₁ t₂ = -1, so t₂ = 2 and Q = (16, 16).
3. Focal distances. The distance of any point (x, y) on the parabola from the focus is a + x. So PS = 4 + 1 = 5 and SQ = 4 + 16 = 20.

   PS : SQ = 5 : 20 = 1 : 4

4. So. m = 1 and n = 4 with gcd(1, 4) = 1, so m² + n² = 1 + 16 = 17, option (a).

The idea to keep. For a focal chord of y² = 4ax, use t₁ t₂ = -1 to get the other end. Then measure lengths from the focus as a + x, with no distance formula needed.

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