Two curves C1: y = x² - 3 and C2: y = k*x² (k is a real number) intersect at two distinct points. The tangent drawn to C2 at one of the intersection points A = (a, y1) where a 0, meets C1 again at point B = (1, y2) with y1 not equal to y2. Find the value of a.

Question

Two curves C1: y = x² - 3 and C2: y = k*x² (k is a real number) intersect at two distinct points. The tangent drawn to C2 at one of the intersection points A = (a, y1) where a > 0, meets C1 again at point B = (1, y2) with y1 not equal to y2. Find the value of a.

Accepted Answer

2. From the intersection condition, k = (a²-3)/a² = 1 - 3/a². The tangent to C2 at A has slope 2ka. Passing through B=(1,-2) on C1: -2 - ka² = 2ka*(1-a). Substituting k and simplifying yields a quadratic in a with solution a=2.

Two curves C1: y = x² - 3 and C2: y = k*x² (k is a real number) intersect at two distinct points. The tangent drawn to C2 at one of the intersection points A = (a, y1) where a > 0, meets C1 again at point B = (1, y2) with y1 not equal to y2. Find the value of a.

Solution

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