To solve the problem step-by-step, we will find the coordinates of point A given that it is at a distance of \(\sqrt{10}\) units from the point (4, 3) and that its ordinate (y-coordinate) is twice its abscissa (x-coordinate).
### Step 1: Define the coordinates of point A
Let the coordinates of point A be \((x, y)\). According to the problem, the ordinate is twice the abscissa, which gives us:
\[ y = 2x \]
### Step 2: Use the distance formula
The distance \(D\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, we want the distance from point A \((x, y)\) to point (4, 3) to be \(\sqrt{10}\):
\[ \sqrt{(x - 4)^2 + (y - 3)^2} = \sqrt{10} \]
### Step 3: Square both sides
To eliminate the square root, we square both sides:
\[ (x - 4)^2 + (y - 3)^2 = 10 \]
### Step 4: Substitute \(y\) with \(2x\)
Now, substitute \(y\) in the equation:
\[ (x - 4)^2 + (2x - 3)^2 = 10 \]
### Step 5: Expand the equation
Now we expand both squares:
1. \((x - 4)^2 = x^2 - 8x + 16\)
2. \((2x - 3)^2 = 4x^2 - 12x + 9\)
Substituting these into the equation gives:
\[ x^2 - 8x + 16 + 4x^2 - 12x + 9 = 10 \]
### Step 6: Combine like terms
Combine all the terms:
\[ 5x^2 - 20x + 25 = 10 \]
### Step 7: Rearrange the equation
Now, rearranging gives:
\[ 5x^2 - 20x + 15 = 0 \]
### Step 8: Simplify the equation
Dividing the entire equation by 5 simplifies it to:
\[ x^2 - 4x + 3 = 0 \]
### Step 9: Factor the quadratic equation
Now, we factor the quadratic:
\[ (x - 3)(x - 1) = 0 \]
### Step 10: Solve for \(x\)
Setting each factor to zero gives us:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x - 1 = 0 \Rightarrow x = 1\)
### Step 11: Find corresponding \(y\) values
Now we can find the corresponding \(y\) values using \(y = 2x\):
1. If \(x = 3\), then \(y = 2(3) = 6\) → Point A is \((3, 6)\)
2. If \(x = 1\), then \(y = 2(1) = 2\) → Point A is \((1, 2)\)
### Final Answer
The coordinates of point A are \((3, 6)\) and \((1, 2)\).
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