Find the point on the x-axis whose distances from the points A(7, 6) and B(-3, 4) are in the ratio 1 : 2.

Answer

To find the point on the x-axis whose distances from the points A(7, 6) and B(-3, 4) are in the ratio 1:2, we can follow these steps:

### Step 1: Define the Point on the X-axis
Let the point on the x-axis be \( P(x, 0) \).

### Step 2: Use the Distance Formula
The distance \( AP \) from point A(7, 6) to point P(x, 0) is given by:
\[ AP = \sqrt{(x - 7)^2 + (0 - 6)^2} = \sqrt{(x - 7)^2 + 36} \]

The distance \( BP \) from point B(-3, 4) to point P(x, 0) is given by:
\[ BP = \sqrt{(x + 3)^2 + (0 - 4)^2} = \sqrt{(x + 3)^2 + 16} \]

### Step 3: Set Up the Ratio
According to the problem, the distances are in the ratio 1:2. Thus, we can write:
\[ \frac{AP}{BP} = \frac{1}{2} \]

Substituting the expressions for \( AP \) and \( BP \):
\[ \frac{\sqrt{(x - 7)^2 + 36}}{\sqrt{(x + 3)^2 + 16}} = \frac{1}{2} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 2\sqrt{(x - 7)^2 + 36} = \sqrt{(x + 3)^2 + 16} \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square roots:
\[ 4((x - 7)^2 + 36) = (x + 3)^2 + 16 \]

### Step 6: Expand Both Sides
Expanding both sides:
\[ 4((x - 7)^2) + 144 = (x^2 + 6x + 9) + 16 \]
\[ 4(x^2 - 14x + 49) + 144 = x^2 + 6x + 25 \]
\[ 4x^2 - 56x + 196 + 144 = x^2 + 6x + 25 \]
\[ 4x^2 - 56x + 340 = x^2 + 6x + 25 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 4x^2 - x^2 - 56x - 6x + 340 - 25 = 0 \]
\[ 3x^2 - 62x + 315 = 0 \]

### Step 8: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3 \), \( b = -62 \), and \( c = 315 \):
\[ x = \frac{62 \pm \sqrt{(-62)^2 - 4 \cdot 3 \cdot 315}}{2 \cdot 3} \]
\[ x = \frac{62 \pm \sqrt{3844 - 3780}}{6} \]
\[ x = \frac{62 \pm \sqrt{64}}{6} \]
\[ x = \frac{62 \pm 8}{6} \]

### Step 9: Calculate the Two Possible Values of x
Calculating the two possible values:
1. \( x = \frac{70}{6} = \frac{35}{3} \)
2. \( x = \frac{54}{6} = 9 \)

### Step 10: Conclusion
Thus, the points on the x-axis are:
1. \( \left(\frac{35}{3}, 0\right) \)
2. \( (9, 0) \)