If the point R divides the line segment joining the point (2, 3) and `(2 tan theta, 3 sec theta), 0 lt theta lt (pi)/(2)` internally in the ratio 2 :3 , then the locus of R is

Options

A. an ellipse length of whose major axis is 12.
B. an ellipse length of whose major axis is 8.
C. a hyperbola length of whose transverse axis is 12.
D. a hyperbola length of whose transverse axis is 8

Answer

To find the locus of the point R that divides the line segment joining the points (2, 3) and (2 tan θ, 3 sec θ) in the ratio 2:3, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and Ratio**:
- Let point A = (2, 3) and point B = (2 tan θ, 3 sec θ).
- The ratio in which R divides AB is 2:3.

2. **Use the Section Formula**:
The coordinates of point R (h, k) that divides the line segment joining A and B in the ratio m:n (here m=2 and n=3) can be calculated using the section formula:
\[ R\left(h, k\right) = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right) \]
where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (2 \tan \theta, 3 \sec \theta) \).

3. **Calculate the x-coordinate (h)**:
\[ h = \frac{2(2 \tan \theta) + 3(2)}{2 + 3} = \frac{4 \tan \theta + 6}{5} \]

4. **Calculate the y-coordinate (k)**:
\[ k = \frac{2(3 \sec \theta) + 3(3)}{2 + 3} = \frac{6 \sec \theta + 9}{5} \]

5. **Express tan θ and sec θ in terms of h and k**:
From the equation for h:
\[ 4 \tan \theta = 5h - 6 \implies \tan \theta = \frac{5h - 6}{4} \]
From the equation for k:
\[ 6 \sec \theta = 5k - 9 \implies \sec \theta = \frac{5k - 9}{6} \]

6. **Use the identity \( \sec^2 \theta = 1 + \tan^2 \theta \)**:
Substitute the expressions for \( \tan \theta \) and \( \sec \theta \):
\[ \sec^2 \theta = \left(\frac{5k - 9}{6}\right)^2 \]
\[ \tan^2 \theta = \left(\frac{5h - 6}{4}\right)^2 \]
Thus,
\[ \left(\frac{5k - 9}{6}\right)^2 = 1 + \left(\frac{5h - 6}{4}\right)^2 \]

7. **Clear the fractions and simplify**:
Multiply through by \( 36 \) to eliminate denominators:
\[ (5k - 9)^2 = 36 + 9(5h - 6)^2 \]
Expand both sides and simplify to get the equation of the locus.

8. **Final Locus Equation**:
After simplification, you will arrive at the equation of a hyperbola.