A tangent line L is drawn at the point (2, -4) on the parabola y² = 8x. If the line L is also tangent to the circle x² + y² = a, then 'a' is equal to ___________.

Solution

Maths sir · 4 steps

Write the tangent at (2, −4), then its distance from the origin is the radius of the circle.

1. Tangent at the point. For y² = 4Ax the tangent at (x₁, y₁) is yy₁ = 2A(x + x₁). Here 4A = 8, so A = 2 and 2A = 4.

   -4y = 4(x + 2) ⇒ x + y + 2 = 0

2. Touching the circle. The circle x² + y² = a has centre (0, 0) and radius √a. A line touches it exactly when its distance from the centre equals the radius.
3. Distance from origin. The distance of x + y + 2 = 0 from (0, 0) is |2|/√(1² + 1²) = 2/√2 = √2. So √a = √2.
4. So. a = (√2)² = 2.

The common slip: 3.2. Writing the tangent as yy₁ = 8(x + x₁) instead of yy₁ = 4(x + x₁) gives the line 2x + y + 4 = 0 and a = 16/5 = 3.2, not 2.

The idea to keep. A line is tangent to a circle when the perpendicular distance from the centre equals the radius. Remember that in x² + y² = a the radius is √a, not a.

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