MHTCET 16 Apr 2026 Shift 1 Paper

Concept:For a line to be a tangent to a hyperbola, the quadratic equation obtained after substitution must have equal roots, meaning its discriminant is zero.Explanation:The hyperbola is $36x^2 - 25y^2 = 3600$.The given line is $y = 2x + \lambda$.Substitute the line into the hyperbola:$$36x^2 - 25(2x+\lambda)^2 = 3600.$$Expanding and simplifying:$$36x^2 -25(4x^2 +4\lambda x + \lambda^2) -3600 =0.$$$$36x^2 -100x^2 -100\lambda x -25\lambda^2 -3600=0.$$$$-64x^2 -100\lambda x -25\lambda^2 -3600=0.$$Multiplying throughout by $-1$:$$64x^2 +100\lambda x +25\lambda^2 +3600=0.$$For tangency, the discriminant must be zero:$$(100\lambda)^2 -4(64)(25\lambda^2+3600)=0.$$$$10000\lambda^2 -6400\lambda^2 -921600=0.$$$$3600\lambda^2 = 921600.$$$$\lambda^2 = 256.$$$$\lambda = \pm 16.$$Hence, option C is correct.Answer:$\lambda = \pm 16$, i.e., Option C.