The line y = x + 1 meets the ellipse x²/4 + y²/2 = 1 at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)² is equal to :

Solution

Maths sir · 4 steps

Substitute the line into the ellipse, then use sum and product of roots to get PQ without finding P and Q.

1. Substitute the line. Put y = x + 1 into the ellipse and multiply through by 4: x² + 2(x+1)² = 4, which gives 3x² + 4x - 2 = 0. Its roots x₁, x₂ are the x-coordinates of P and Q.
2. Sum and product. x₁ + x₂ = -4/3 and x₁ x₂ = -2/3. So (x₁ - x₂)² = 16/9 + 8/3 = 40/9.
3. Length of PQ. The line has slope 1, so y₁ - y₂ = x₁ - x₂. The distance formula then gives the chord length.

   PQ² = (1 + m²)(x₁ - x₂)² = 2 × 40/9 = 80/9

4. So. PQ is a diameter, so r² = PQ²/4 = 20/9. Then (3r)² = 9r² = 20, which is option (a).

The idea to keep. For a chord cut by a line of slope m, length squared is (1+m²)(x₁-x₂)². Get (x₁-x₂)² from sum and product of roots. There is no need to solve the quadratic.

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