18. The straight lines joining the origin to the points of intersection of the line `4x + 3y = 24` with the curve `(x - 3)^2 + (y - 4)^2 = 25 :`

Options

A. are coincident
B. are perpendicular
C. make equal angles with x-axis
D. none of these

Answer

To solve the problem of finding the points of intersection of the line \(4x + 3y = 24\) with the circle \((x - 3)^2 + (y - 4)^2 = 25\), we will follow these steps:

### Step 1: Write down the equations
We have the line:
\[ 4x + 3y = 24 \]
And the circle:
\[ (x - 3)^2 + (y - 4)^2 = 25 \]

### Step 2: Express \(y\) in terms of \(x\) from the line equation
From the line equation, we can express \(y\) in terms of \(x\):
\[ 3y = 24 - 4x \implies y = \frac{24 - 4x}{3} \]

### Step 3: Substitute \(y\) in the circle equation
Now, substitute \(y\) into the circle equation:
\[ (x - 3)^2 + \left(\frac{24 - 4x}{3} - 4\right)^2 = 25 \]

### Step 4: Simplify the equation
First, simplify the term \(\frac{24 - 4x}{3} - 4\):
\[ \frac{24 - 4x}{3} - 4 = \frac{24 - 4x - 12}{3} = \frac{12 - 4x}{3} \]
Now substitute this back into the circle equation:
\[ (x - 3)^2 + \left(\frac{12 - 4x}{3}\right)^2 = 25 \]

### Step 5: Expand both sides
Expanding the left-hand side:
\[ (x - 3)^2 = x^2 - 6x + 9 \]
And,
\[ \left(\frac{12 - 4x}{3}\right)^2 = \frac{(12 - 4x)^2}{9} = \frac{144 - 96x + 16x^2}{9} \]
So we have:
\[ x^2 - 6x + 9 + \frac{144 - 96x + 16x^2}{9} = 25 \]

### Step 6: Clear the fraction
Multiply through by 9 to eliminate the fraction:
\[ 9(x^2 - 6x + 9) + (144 - 96x + 16x^2) = 225 \]
This simplifies to:
\[ 9x^2 - 54x + 81 + 144 - 96x + 16x^2 = 225 \]
Combining like terms:
\[ (9x^2 + 16x^2) + (-54x - 96x) + (81 + 144 - 225) = 0 \]
\[ 25x^2 - 150x = 0 \]

### Step 7: Factor the quadratic equation
Factoring out \(25x\):
\[ 25x(x - 6) = 0 \]
Thus, \(x = 0\) or \(x = 6\).

### Step 8: Find corresponding \(y\) values
For \(x = 0\):
\[ y = \frac{24 - 4(0)}{3} = 8 \implies (0, 8) \]
For \(x = 6\):
\[ y = \frac{24 - 4(6)}{3} = 0 \implies (6, 0) \]

### Step 9: Conclusion
The points of intersection are \((0, 8)\) and \((6, 0)\).