If y = m₁x + c₁ and y = m₂x + c₂, m₁ ≠ m₂ are two common tangents of circle x² + y² = 2 and parabola y² = x, then the value of 8|m₁m₂| is equal to :

Solution

Maths sir · 4 steps

Write the parabola's tangent in slope form, then make its distance from the centre equal the radius.

1. Tangent to parabola. Compare y² = x with y² = 4ax to get a = 1/4. Every tangent with slope m is y = mx + a/m, that is y = mx + 1/4m.
2. Touch the circle. The circle has centre (0, 0) and radius √2. The line y = mx + c touches it when its distance from the centre is √2, so c² = 2(1 + m²). Put c = 1/4m, so 1/16m² = 2 + 2m². Multiply both sides by 16m².

   32m⁴ + 32m² - 1 = 0

3. Solve for m squared. As a quadratic in m²: m² = (-32 ± √(1024 + 128))/64 = (-32 ± 24√2)/64. Only the plus sign gives a positive value, so m² = (-4 + 3√2)/8.
4. So. The two slopes are m₁ = m and m₂ = -m, so |m₁ m₂| = m². Then 8|m₁ m₂| = -4 + 3√2, which is option (c).

The idea to keep. For a common tangent, write the tangent to one curve in slope form, then apply the other curve's tangency condition. The roots of the equation you get in m are the slopes of the common tangents.

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