Find the coordinates of the point of intersection of tangent at the points where `x+ 4y - 14 =0 ` meets the circle ` x^(2) + y^(2) - 2x+ 3y -5=0 `

Answer

To find the coordinates of the point of intersection of the tangents at the points where the line \( x + 4y - 14 = 0 \) meets the circle \( x^2 + y^2 - 2x + 3y - 5 = 0 \), we can follow these steps:

### Step 1: Identify the equations
We have the line:
\[ L: x + 4y - 14 = 0 \]
And the circle:
\[ C: x^2 + y^2 - 2x + 3y - 5 = 0 \]

### Step 2: Find the points of intersection
To find the points of intersection of the line and the circle, we can substitute \( y \) from the line equation into the circle equation.

From the line equation, we can express \( y \):
\[ 4y = 14 - x \implies y = \frac{14 - x}{4} \]

Now substitute \( y \) into the circle's equation:
\[ x^2 + \left(\frac{14 - x}{4}\right)^2 - 2x + 3\left(\frac{14 - x}{4}\right) - 5 = 0 \]

### Step 3: Simplify the equation
Substituting \( y \):
\[ x^2 + \frac{(14 - x)^2}{16} - 2x + \frac{3(14 - x)}{4} - 5 = 0 \]
Multiply through by 16 to eliminate the fraction:
\[ 16x^2 + (14 - x)^2 - 32x + 12(14 - x) - 80 = 0 \]
Expanding the terms:
\[ 16x^2 + (196 - 28x + x^2) - 32x + 168 - 12x - 80 = 0 \]
Combine like terms:
\[ 17x^2 - 72x + 284 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{72 \pm \sqrt{(-72)^2 - 4 \cdot 17 \cdot 284}}{2 \cdot 17} \]
Calculating the discriminant:
\[ b^2 - 4ac = 5184 - 19256 = -14072 \]
Since the discriminant is negative, there are no real intersection points, indicating that the line does not intersect the circle.

### Step 5: Find the coordinates of the point of intersection of the tangents
Since the line does not intersect the circle, we can find the equation of the chord of contact from the point where the tangents touch the circle. The equation of the chord of contact from point \( (x_1, y_1) \) is given by:
\[ xx_1 + yy_1 - 2x - 3y - 5 = 0 \]
However, since we need the coordinates of the point of intersection of the tangents, we can use the formula for the intersection of tangents at points where the line meets the circle.

### Step 6: Use the chord of contact
The chord of contact for the points of tangency can be derived from the line equation:
\[ x + 4y - 14 = 0 \]
This gives us the coefficients for the tangents.

### Final Coordinates
To find the coordinates of the point of intersection of the tangents, we can set up the equations derived from the coefficients and solve them.