The ordinates of the points P and Q on the parabola with focus (3,0) and directrix x=-3 are in the ratio 3: 1. If R(α, β) is the point of intersection of the tangents to the parabola at P and Q, then β²/α is equal to _______________.

Solution

Maths sir · 4 steps

Write P and Q in parametric form, use the ratio to link t₁ and t₂, and the parameter cancels.

1. Find the parabola. The focus (3,0) and directrix x=-3 are symmetric about the origin, so the vertex is at the origin and a=3. The parabola is y² = 12x.
2. Use the ratio. Take P(3t₁², 6t₁) and Q(3t₂², 6t₂). The ordinates are in the ratio 3:1, so 6t₁ = 3 × 6t₂, which gives t₁ = 3t₂.

   R = (at₁t₂, a(t₁+t₂)) = (9t₂², 12t₂)

3. Form the ratio. Here α = 3 × 3t₂ × t₂ = 9t₂² and β = 3(3t₂ + t₂) = 12t₂. So β²/α = 144t₂²/9t₂², and t₂ cancels.
4. So. β²/α = 16, whichever points P and Q you pick.

The common slip: 64. Taking the intersection's y-coordinate as 2a(t₁+t₂) instead of a(t₁+t₂) gives β = 24t₂ and the answer 64.

The idea to keep. Tangents at t₁ and t₂ on y²=4ax meet at (at₁t₂, a(t₁+t₂)). Any fixed relation between t₁ and t₂ makes such ratios a constant.

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