The curves C 1 : y = x 2 − 3 and C 2 : y = k x 2 ( k 0 ) meets C 1 again at B = ( 1 , y 2 ) ( y 1  = y 2 ) . Find

Question

The curves C 1 ​ : y = x 2 − 3 and C 2 ​ : y = k x 2 ( k < 1 ) meet at two points. The tangent to C 2 ​ at A = ( a , y 1 ​ ) ( a > 0 ) meets C 1 ​ again at B = ( 1 , y 2 ​ ) ( y 1 ​  = y 2 ​ ) . Find a .

TestHub · Accepted Answer

At A , ( 1 − k ) a 2 = 3 , so k = 1 − 3/ a 2 . The tangent to C 2 ​ is y = 2 ka x − k a 2 . Substituting B = ( 1 , − 2 ) gives a 2 − 2 a − 3 = 0 after excluding a = 1 (same intersection). Therefore a = 3 .

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