(a) Find the coordinates of the point which divides the line joining (7, -5) and (-2, 7) externally in the ration 3 : 2. (b) Without using calculators or ma...

Answer Details

a)

Let the point dividing the line joining (7, -5) and (-2, 7) externally in the ratio 3 : 2 be (x, y).

We can use the section formula to find the coordinates of the point:

x = (2*7 + 3*(-2))/5 = 1

y = (2*(-5) + 3*7)/5 = 1

Therefore, the coordinates of the point are (1, 1).

b)

To simplify the expression \(\frac{2}{1 + \sqrt{2}}\) - \(\frac{2}{2 + \sqrt{2}}\), we need to use the conjugate of the denominator to eliminate the radicals in the denominator.

Notice that the conjugate of \(1 + \sqrt{2}\) is \(1 - \sqrt{2}\), and the conjugate of \(2 + \sqrt{2}\) is \(2 - \sqrt{2}\).

Multiplying the first fraction by \(\frac{2 - \sqrt{2}}{2 - \sqrt{2}}\) and the second fraction by \(\frac{1 - \sqrt{2}}{1 - \sqrt{2}}\), we get:

\(\frac{2(2-\sqrt{2})}{(1+\sqrt{2})(2-\sqrt{2})}-\frac{2(1-\sqrt{2})}{(2+\sqrt{2})(1-\sqrt{2})}\)

Simplifying the numerators and denominators, we get:

\(\frac{4-2\sqrt{2}}{1}-\frac{2-4\sqrt{2}}{1} = 2\sqrt{2}-2\)

Therefore, the answer is in the form p + q\(\sqrt{n}\), where p = -2, q = 2, and n = 2.

The reasoning behind this is that we have a rational number (2) added to an irrational number (\(2\sqrt{2}\)), which gives us an expression in the form p + q\(\sqrt{n}\), where p and q are rational numbers and n is an integer. We can then identify p, q, and n by comparing the coefficients of the rational and irrational parts of the expression.