Equation of pair of tangents to the ellipse `9x^(2)+25y^(2)=225` from a point (4,2) is

Options

A. `9x^(2)+25y^(2)-225=((36x+50y-225)/(sqrt(19)))^(2)`
B. `9x^(2)+25y^(2)-225=((36x+25y-225)/(15sqrt(19)))^(2)`
C. `((x^(2))/(25)+(y^(2))/(9)-1)19=((36x+25y-225)/(15))^(2)`
D. none

Answer

To find the equation of the pair of tangents to the ellipse \(9x^2 + 25y^2 = 225\) from the point \(P(4, 2)\), we can follow these steps:

### Step 1: Write the equation of the ellipse in standard form
The given equation of the ellipse is:
\[ 9x^2 + 25y^2 = 225 \]
Dividing the entire equation by 225, we get:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
This shows that the semi-major axis \(a = 5\) and the semi-minor axis \(b = 3\).

### Step 2: Identify the point from which tangents are drawn
The point from which we are drawing the tangents is \(P(4, 2)\).

### Step 3: Use the formula for the equation of the pair of tangents
The equation of the pair of tangents from a point \((x_1, y_1)\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is given by:
\[ S S_1 = T^2 \]
where:
- \(S = \frac{x^2}{25} + \frac{y^2}{9} - 1\)
- \(S_1 = \frac{x_1^2}{25} + \frac{y_1^2}{9} - 1\)
- \(T = \frac{xx_1}{25} + \frac{yy_1}{9}\)

### Step 4: Calculate \(S_1\)
Substituting \(x_1 = 4\) and \(y_1 = 2\):
\[ S_1 = \frac{4^2}{25} + \frac{2^2}{9} - 1 = \frac{16}{25} + \frac{4}{9} - 1 \]
Finding a common denominator (225):
\[ S_1 = \frac{144}{225} + \frac{100}{225} - \frac{225}{225} = \frac{244 - 225}{225} = \frac{19}{225} \]

### Step 5: Substitute \(S\) and \(S_1\) into the equation
Now, substituting \(S\) and \(S_1\) into the equation \(S S_1 = T^2\):
\[ \left(\frac{x^2}{25} + \frac{y^2}{9} - 1\right) \left(\frac{19}{225}\right) = \left(\frac{4x}{25} + \frac{2y}{9}\right)^2 \]

### Step 6: Simplify the equation
Expanding the left-hand side:
\[ \frac{19}{225} \left(\frac{x^2}{25} + \frac{y^2}{9} - 1\right) = \frac{19x^2}{5625} + \frac{19y^2}{2025} - \frac{19}{225} \]
And the right-hand side:
\[ \left(\frac{4x}{25} + \frac{2y}{9}\right)^2 = \frac{16x^2}{625} + \frac{16xy}{225} + \frac{4y^2}{81} \]

### Step 7: Equate and rearrange
Equating both sides and rearranging gives us the final equation of the pair of tangents.

### Final Step: Write the final equation
After simplification, we find the equation of the pair of tangents to be:
\[ 9x^2 + 25y^2 - 225 = 36x + 50y \]