If the common tangent to the parabolas,
y² = 4x and x² = 4y also touches the circle, x² + y² = c²,
then c is equal to :

Solution

Maths sir · 5 steps

Write the slope form of the tangent for each parabola and match the intercepts. The slope comes out at once.

1. Tangent to first. For y² = 4x we have a = 1, so every tangent with slope m is y = mx + 1/m.
2. Tangent to second. For x² = 4y we have a = 1, and the tangent with slope m is y = mx - m². You can check this by putting y = mx + k into x² = 4y and setting the discriminant of x² - 4mx - 4k = 0 to zero, which gives k = -m².

   1/m = -m² ⇒ m³ = -1 ⇒ m = -1

3. Common tangent. With m = -1 the intercept is 1/m = -1, so the common tangent is y = -x - 1, that is, x + y + 1 = 0.
4. Touch the circle. A line touches x² + y² = c² when its distance from the centre (0, 0) equals the radius. That distance is (|0 + 0 + 1|)/√(1² + 1²) = 1/√2.
5. So. c = 1/√2, which is option (a).

The idea to keep. For a common tangent, write each curve's tangent in slope form and equate the intercepts. For a circle, tangency means the perpendicular distance from the centre equals the radius.

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