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Let a curve E be the locus of the centre of a circle that passes through the point (5, 2) and touches the line x = -1 . If the point of intersection of the lines x - y = 0 and x + y = 6 lies on the curve E , then the possible values of are ₁ and ₂ . If ₁ < ₂ , then the value of 14 ₁ + 10 ₂ is equal to

Correct answer

12

Step-by-step solution

Let the centre of the circle be (h, k) . Since the circle passes through (5, 2) and touches the line x = -1 , its distance from the point (5, 2) is equal to its perpendicular distance from the line x = -1 . (h-5)^2 + (k-2)^2 = |h + 1| Squaring both sides gives (h-5)^2 + (k-2)^2 = (h+1)^2 . h^2 - 10h + 25 + (k-2)^2 = h^2 + 2h + 1 (k-2)^2 = 12(h-2) . Thus, the curve E is the parabola (y-2)^2 = 12(x-2) . The point of intersection of the lines x - y = 0 and x + y = 6 is found by substituting y = x into the second equat

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