The curve y = 7 - \(\frac{6}{x}\) and the line y + 2x - 3 = 0 intersect at two point. Finf the; (a) coordinates of the two points (b) equation of the perpen...

Answer Details

(a)

To find the coordinates of the two points where the curve \(y = 7 - \frac{6}{x}\) intersects the line \(y + 2x - 3 = 0\), we can substitute \(y\) in the equation of the line with the expression for \(y\) in the equation of the curve and solve for \(x\).

\(y + 2x - 3 = 0\) becomes:

\(7 - \frac{6}{x} + 2x - 3 = 0\)

Simplifying this expression, we get:

\(2x^2 - x - 2 = 0\)

We can factorize this quadratic equation to get:

\((2x + 1)(x - 2) = 0\)

This gives us two solutions for \(x\): \(x = -\frac{1}{2}\) and \(x = 2\).

We can substitute these values of \(x\) into either equation to get the corresponding values of \(y\).

For \(x = -\frac{1}{2}\):

\(y + 2(-\frac{1}{2}) - 3 = 0\)

\(y = 4\)

So one point of intersection is \(-\frac{1}{2}, 4\).

For \(x = 2\):

\(y + 2(2) - 3 = 0\)

\(y = -1\)

So the other point of intersection is \(2, -1\).

Therefore, the coordinates of the two points of intersection are \(-\frac{1}{2}, 4\) and \(2, -1\).

(b)

To find the equation of the perpendicular bisector of the line joining the two points of intersection, we first need to find the midpoint of the line segment joining the two points.

The midpoint formula gives us:

\(\left(\frac{x_1 + x_2}{2}\), \(\frac{y_1 + y_2}{2}\right)\)

Substituting the values of the two points, we get:

\(\left(\frac{-\frac{1}{2} + 2}{2}\), \(\frac{4 - 1}{2}\right)\)

Simplifying this expression, we get the midpoint \(\left(\frac{3}{4}, \frac{3}{2}\right)\).

Next, we need to find the slope of the line joining the two points.

The slope formula gives us:

\(\frac{y_2 - y_1}{x_2 - x_1}\)

Substituting the values of the two points, we get:

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

Simplifying this expression, we get the slope \(-\frac{10}{9}\).

The perpendicular bisector of the line segment joining the two points will have a slope that is the negative reciprocal of this slope, which is \(\frac{9}{10}\).

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