Consider the parabola whose focus is at (0,0) and tangent at vertex is x-y+1=0
Tangents drawn to the parabola at the extremities of the chord 3x+2y=0 intersect at what angle ?

Options

A. `pi//6`
B. `pi//3`
C. `pi//2`
D. none of these

Explanation

To solve the problem, we need to find the angle at which the tangents drawn to the parabola at the extremities of the chord \(3x + 2y = 0\) intersect. The parabola has its focus at the origin (0,0) and a tangent at the vertex given by the equation \(x - y + 1 = 0\).

### Step-by-Step Solution:

1. **Identify the Parabola**:
The focus of the parabola is at (0,0) and the tangent at the vertex is given by \(x - y + 1 = 0\). This line can be rewritten as \(y = x + 1\).

2. **Find the Directrix**:
The distance from the focus to the tangent line can be calculated using the formula for the distance from a point to a line:
\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = -1\), \(C = 1\), and the point is (0,0):
\[ \text{Distance} = \frac{|1 \cdot 0 - 1 \cdot 0 + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}} \]
The directrix is parallel to the tangent and located at a distance of \(2 \times \frac{1}{\sqrt{2}} = \sqrt{2}\) from the focus. Thus, the equation of the directrix is:
\[ x - y + k = 0 \quad \text{where } k = -2 \]
Therefore, the directrix is \(x - y - 2 = 0\).

3. **Equation of the Parabola**:
The equation of the parabola can be derived from the definition of a parabola: the distance from any point on the parabola to the focus is equal to the distance to the directrix. The general form of the parabola is:
\[ y^2 = 4px \]
Here, \(p\) is the distance from the focus to the directrix. Since the directrix is at \(y = x - 2\), we can derive the equation of the parabola.

4. **Find the Points of Intersection with the Chord**:
The chord given is \(3x + 2y = 0\) or \(y = -\frac{3}{2}x\). We can substitute this into the parabola's equation to find the points of intersection.

5. **Find the Tangents at Extremities**:
The tangents at the points of intersection can be found using the point-slope form of the tangent line equation.

6. **Calculate the Angle Between the Tangents**:
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
If the tangents are perpendicular, then \(m_1 m_2 = -1\), which implies that the angle between them is \(90^\circ\) or \(\frac{\pi}{2}\).

### Conclusion:
The tangents drawn to the parabola at the extremities of the chord \(3x + 2y = 0\) intersect at an angle of \(90^\circ\) (or \(\frac{\pi}{2}\)).