If `I_(1),I_(2),I_(3)` are the intercept on x-axis, y-axis, y=x w.r.t `x^(2)+y^(2)-14x-10y+24=0` then

Options

A. `I_(2)gtI_(3)gtI_(1)`
B. `I_(3)gtI_(1)gtI_(2)`
C. `I_(1)gtI_(3)gtI_(2)`
D. `I_(3)gtI_(2)gtI_(1)`

Answer

To solve the problem, we need to find the intercepts \( I_1, I_2, I_3 \) of the circle given by the equation \( x^2 + y^2 - 14x - 10y + 24 = 0 \) on the x-axis, y-axis, and the line \( y = x \) respectively.

### Step 1: Rewrite the Circle Equation
First, we rewrite the given equation of the circle in a more standard form by completing the square.

The equation is:
\[ x^2 + y^2 - 14x - 10y + 24 = 0 \]

Completing the square for \( x \) and \( y \):
- For \( x \):
\[ x^2 - 14x = (x - 7)^2 - 49 \]
- For \( y \):
\[ y^2 - 10y = (y - 5)^2 - 25 \]

Substituting back into the equation:
\[ (x - 7)^2 - 49 + (y - 5)^2 - 25 + 24 = 0 \]
\[ (x - 7)^2 + (y - 5)^2 - 50 = 0 \]
\[ (x - 7)^2 + (y - 5)^2 = 50 \]

This represents a circle with center \( (7, 5) \) and radius \( \sqrt{50} \).

### Step 2: Find the x-intercept \( I_1 \)
To find the x-intercept, set \( y = 0 \):
\[ (x - 7)^2 + (0 - 5)^2 = 50 \]
\[ (x - 7)^2 + 25 = 50 \]
\[ (x - 7)^2 = 25 \]
Taking the square root:
\[ x - 7 = 5 \quad \text{or} \quad x - 7 = -5 \]
Thus, \( x = 12 \) or \( x = 2 \).

The x-intercepts are \( 12 \) and \( 2 \). Therefore, the length of the x-intercept \( I_1 \) is:
\[ I_1 = 12 - 2 = 10 \]

### Step 3: Find the y-intercept \( I_2 \)
To find the y-intercept, set \( x = 0 \):
\[ (0 - 7)^2 + (y - 5)^2 = 50 \]
\[ 49 + (y - 5)^2 = 50 \]
\[ (y - 5)^2 = 1 \]
Taking the square root:
\[ y - 5 = 1 \quad \text{or} \quad y - 5 = -1 \]
Thus, \( y = 6 \) or \( y = 4 \).

The y-intercepts are \( 6 \) and \( 4 \). Therefore, the length of the y-intercept \( I_2 \) is:
\[ I_2 = 6 - 4 = 2 \]

### Step 4: Find the intercept \( I_3 \) on the line \( y = x \)
Substituting \( y = x \) into the circle's equation:
\[ (x - 7)^2 + (x - 5)^2 = 50 \]
Expanding:
\[ (x - 7)^2 + (x - 5)^2 = (x^2 - 14x + 49) + (x^2 - 10x + 25) = 50 \]
Combining like terms:
\[ 2x^2 - 24x + 74 = 50 \]
\[ 2x^2 - 24x + 24 = 0 \]
Dividing by 2:
\[ x^2 - 12x + 12 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{12 \pm \sqrt{144 - 48}}{2} = \frac{12 \pm \sqrt{96}}{2} = 6 \pm 4\sqrt{6} \]
The intercepts are \( 6 + 4\sqrt{6} \) and \( 6 - 4\sqrt{6} \).

Thus, the length of the intercept \( I_3 \) is:
\[ I_3 = (6 + 4\sqrt{6}) - (6 - 4\sqrt{6}) = 8\sqrt{6} \]

### Step 5: Compare \( I_1, I_2, I_3 \)
Now we have:
- \( I_1 = 10 \)
- \( I_2 = 2 \)
- \( I_3 = 8\sqrt{6} \)

Calculating \( 8\sqrt{6} \):
\[ \sqrt{6} \approx 2.45 \quad \Rightarrow \quad 8\sqrt{6} \approx 19.6 \]

Thus, we find:
\[ I_3 > I_1 > I_2 \quad \Rightarrow \quad 8\sqrt{6} > 10 > 2 \]

### Conclusion
The correct order of the intercepts is:
\[ I_3 > I_1 > I_2 \]