The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` forms a triangle of constant area with the coordinate axes is (a) straight line (b) a hyperbola (c) an ellipse (d) a circle

Options

A. a straight line
B. a hyperbola
C. an ellipse
D. a circle

Explanation

To solve the problem, we need to find the locus of a point from which the chord of contact of tangents drawn to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) forms a triangle of constant area with the coordinate axes.

### Step-by-step Solution:

1. **Equation of the Chord of Contact:**
The equation of the chord of contact of tangents drawn from a point \((h, k)\) to the ellipse is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Substituting \(x_1 = h\) and \(y_1 = k\), we have:
\[ \frac{xh}{a^2} + \frac{yk}{b^2} = 1 \]

2. **Finding Intercepts:**
To find the intercepts on the x-axis and y-axis:
- For the x-intercept (where \(y = 0\)):
\[ \frac{xh}{a^2} = 1 \implies x = \frac{a^2}{h} \]
- For the y-intercept (where \(x = 0\)):
\[ \frac{yk}{b^2} = 1 \implies y = \frac{b^2}{k} \]

3. **Coordinates of the Intercepts:**
The intercepts on the axes are:
- \(A\left(\frac{a^2}{h}, 0\right)\) on the x-axis
- \(B\left(0, \frac{b^2}{k}\right)\) on the y-axis

4. **Area of the Triangle Formed:**
The area \(A\) of the triangle formed by these intercepts and the origin \((0, 0)\) is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{a^2}{h} \times \frac{b^2}{k} \]
Therefore,
\[ A = \frac{a^2 b^2}{2hk} \]

5. **Constant Area Condition:**
Since the area is constant, we can set:
\[ \frac{a^2 b^2}{2hk} = C \quad \text{(where \(C\) is a constant)} \]
Rearranging gives:
\[ hk = \frac{a^2 b^2}{2C} \]
Let \(k = \frac{a^2 b^2}{2C h}\).

6. **Identifying the Locus:**
The equation \(hk = k_0\) (where \(k_0 = \frac{a^2 b^2}{2C}\)) represents a rectangular hyperbola. Thus, the locus of the point \((h, k)\) is a hyperbola.

### Conclusion:
The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse forms a triangle of constant area with the coordinate axes is a hyperbola.