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.
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
- 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:
$$ x = \frac{1 \cdot \left( -\frac{4}{3} \right) + 2 \cdot \frac{7}{2}}{1 + 2} $$
$$ y = \frac{1 \cdot \left( \frac{1}{3} \right) + 2 \cdot (-5)}{1 + 2} $$
- Calculating ( x ) Coordinate
Calculating the ( x )-coordinate gives:
$$ x = \frac{-\frac{4}{3} + 7}{3} = \frac{-\frac{4}{3} + \frac{21}{3}}{3} = \frac{\frac{17}{3}}{3} = \frac{17}{9} $$
- Calculating ( y ) Coordinate
Now, calculating the ( y )-coordinate:
$$ y = \frac{\frac{1}{3} - 10}{3} = \frac{\frac{1}{3} - \frac{30}{3}}{3} = \frac{-\frac{29}{3}}{3} = -\frac{29}{9} $$
- 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).
- 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.
- 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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