JEEOlympiad
A single straight line is a common tangent to both parabolas y^2 = 4x…
Question
A single straight line is a common tangent to both parabolas y^2 = 4x and x^2 = 4y. It touches the first parabola at the point P and the second at the point Q. Find the square of the distance PQ.
✓ Verified answer: 18checked by our engine — not a guess
Step-by-step solution
Let the common tangent be y = mx + c.
Tangency to y^2 = 4x: substituting gives (mx+c)^2 = 4x, i.e.
m^2 x^2 + (2mc-4)x + c^2 = 0; the discriminant must vanish: (2mc-4)^2 - 4 m^2 c^2 = 0 => -16mc + 16 = 0 => mc = 1, so c = 1/m.
Tangency to x^2 = 4y: x^2 = 4(mx+c) => x^2 - 4mx - 4c = 0; discriminant zero: 16 m^2 + 16 c = 0 => c = -m^2.
Setting 1/m = -m^2 gives m^3 = -1 => m = -1, hence c = -1. The common tangent is y = -x - 1.
Point of contact on y^2 = 4x: double root x = -(2mc-4)/(2m^2) = -(2(-1)(-1)-4)/2 = -(2-4)/2 = 1, y = -1-1 = -2, so P = (1, -2).
Point of contact on x^2 = 4y: x = 2m = -2, y = (-1)(-2)-1 = 1, so Q = (-2, 1).
PQ^2 = (1-(-2))^2 + (-2-1)^2 = 9 + 9 = 18.
Final answer18
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