a) Find the coordinates of the point that divides the line segment joining points A(7,-5) and B(2,3) into the ratio 1:2. b) Sketch the graph of the equation 9x^2 + 4y^2 = 25. c) Sh... a) Find the coordinates of the point that divides the line segment joining points A(7,-5) and B(2,3) into the ratio 1:2. b) Sketch the graph of the equation 9x^2 + 4y^2 = 25. c) Show that the distance d from a point P(x1, y1) to a line L with equation ax + by + c = 0 is given by the formula d = |ax1 + by1 + c| / sqrt(a^2 + b^2). d) Analyze the equation 16x^2 + 9y^2 + 64x - 18y - 71 = 0. Read more

Steps to Solve

1. Finding the Coordinates of the Point

To find the coordinates of the point that divides the line segment joining points ( A \left( \frac{7}{2}, -5 \right) ) and ( B \left( -\frac{4}{3}, \frac{1}{3} \right) ) in the ratio ( m:n ), we use the section formula:

$$ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$

Let ( m = 1 ) and ( n = 2 ) (since the ratio is 1:2).

Coordinates of ( A ): ( x_1 = \frac{7}{2}, y_1 = -5 )

Coordinates of ( B ): ( x_2 = -\frac{4}{3}, y_2 = \frac{1}{3} )

Now substituting these values, we find:

$$ x = \frac{1 \cdot \left( -\frac{4}{3} \right) + 2 \cdot \frac{7}{2}}{1 + 2} $$

$$ y = \frac{1 \cdot \left( \frac{1}{3} \right) + 2 \cdot (-5)}{1 + 2} $$

2. Calculating ( x ) Coordinate

Calculating the ( x )-coordinate gives:

$$ x = \frac{-\frac{4}{3} + 7}{3} = \frac{-\frac{4}{3} + \frac{21}{3}}{3} = \frac{\frac{17}{3}}{3} = \frac{17}{9} $$

3. Calculating ( y ) Coordinate

Now, calculating the ( y )-coordinate:

$$ y = \frac{\frac{1}{3} - 10}{3} = \frac{\frac{1}{3} - \frac{30}{3}}{3} = \frac{-\frac{29}{3}}{3} = -\frac{29}{9} $$

4. Graphing the Equation

To sketch the graph of the equation ( 9x^2 + 4y^2 = 25 ), rewrite it in standard form:

$$ \frac{x^2}{\left( \frac{5}{3} \right)^2} + \frac{y^2}{\left( \frac{5}{2} \right)^2} = 1 $$

This is an ellipse centered at the origin, where the semi-major axis is ( \frac{5}{2} ) (along the y-axis) and the semi-minor axis is ( \frac{5}{3} ) (along the x-axis).

5. Showing the Distance Formula

The distance ( d ) from a point ( P(x_1, y_1) ) to a line given by the equation ( ax + by + c = 0 ) is given by:

$$ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} $$

For the line ( ax + by + c = 0 ) where ( a = 1, b = 1, c = -1 ), substitute ( x_1, y_1 ) into the formula to show it's valid.

6. Analyzing the Quadratic Equation

To analyze the equation ( 16x^2 + 9y^2 + 64x - 18y - 71 = 0 ), we would complete the square for both ( x ) and ( y ) terms to put it into standard form.