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.

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Understand the Problem

The question involves several mathematical tasks, including finding points on a line, sketching a graph, showing a distance formula, and analyzing a quadratic equation.

Answer

The coordinates of the point are \( \left( \frac{17}{9}, -\frac{29}{9} \right) \).
Answer for screen readers

The coordinates of the point are ( \left( \frac{17}{9}, -\frac{29}{9} \right) ).

For the graph equation ( 9x^2 + 4y^2 = 25 ), it's an ellipse centered at the origin with semi-major axis ( \frac{5}{2} ) and semi-minor axis ( \frac{5}{3} ).

For the distance from a point ( P ) to the line, the formula is given as:

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

The analysis of the quadratic will depend on completing the square.

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:

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

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

  1. Calculating ( x ) Coordinate

Calculating the ( x )-coordinate gives:

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

  1. Calculating ( y ) Coordinate

Now, calculating the ( y )-coordinate:

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

  1. 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).

  1. 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.

  1. 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.

The coordinates of the point are ( \left( \frac{17}{9}, -\frac{29}{9} \right) ).

For the graph equation ( 9x^2 + 4y^2 = 25 ), it's an ellipse centered at the origin with semi-major axis ( \frac{5}{2} ) and semi-minor axis ( \frac{5}{3} ).

For the distance from a point ( P ) to the line, the formula is given as:

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

The analysis of the quadratic will depend on completing the square.

More Information

The point dividing the segment represents a weighted average based on the ratio of division, and the ellipse depicts a geometric shape that can arise from quadratic equations in two variables. The distance formula is a fundamental result in geometry which provides a method for calculating the shortest distance from a point to a line.

Tips

  • Misapplying the section formula: Ensure to use the right coordinates in the correct positions.
  • Forgetting to simplify expressions: Always check and simplify where possible.
  • Confusion in graphing: Be careful with axis lengths and orientations when sketching ellipses.

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