Question

The equation of a curve is y^2 mx^2 = m, where m is a positive constant. Find the largest integer value of m for which the line x-y=3 does not meet the curve.

Answer 1

The largest integer value of m for which the line x-y=3 does not meet the curve y^2 = mx^2 + m is 6.

Step-by-step explanation:

The equation of the curve is y^2 = mx^2 + m, where m is a positive constant. To find the largest integer value of m for which the line x-y=3 does not meet the curve, we need to substitute the equation of the line into the equation of the curve and check for any values of m that make the equation impossible to satisfy.

1. Substitute x - y = 3 into the curve equation: (x - y)^2 = mx^2 + m
2. Simplify the equation: (x^2 - 2xy + y^2) = mx^2 + m
3. Since we are looking for integer values of m, test values of m starting from the largest positive integer and work downwards until we find a value that satisfies the equation. If the equation cannot be satisfied, move to the next lower integer value of m.
4. By substituting m = 6 into the equation, we find a solution. Therefore, the largest integer value of m for which the line does not meet the curve is 6.