Question

Let the eccentricity of an ellipse be $\frac{1}{2}$. If $S(3,2)$ is a focus and $x-9=0$ is the corresponding directrix of the ellipse, then the equation of the ellipse is}

• A. $3x^{2}+4y^{2}-6x-16y-29=0$
• B. $3x^{2}+4y^{2}-8x-16y-29=0$
• C. $3x^{2}+4y^{2}-6x-12y-29=0$
• D. $3x^{2}+4y^{2}-6x-16y+29=0$
• E. $13x^{2}+5y^{2}-6x-16y-29=0$

Hint:

Math Tip: The general definition of a conic ($SP = e \cdot PM$) is incredibly powerful because it applies universally to parabolas ($e=1$), ellipses ($e<1$), and hyperbolas ($e>1$) without needing to memorize specific standard-form variations.

Answer 1

The Correct Option is A

Concept:
Coordinate Geometry - Conic Sections Definition.
For any point $P(x,y)$ on a conic, the ratio of its distance from the focus $S$ to its perpendicular distance from the directrix $M$ is equal to the eccentricity $e$. $$ SP = e \cdot PM \implies SP^2 = e^2 \cdot PM^2 $$
Step 1: Identify the given parameters.

• Focus $S = (3, 2)$
• Eccentricity $e = \frac{1}{2}$
• Directrix line: $x - 9 = 0$

Step 2: Set up the distance equation.
Let $P(x, y)$ be any point on the ellipse.

• Distance $SP$ squared: $(x - 3)^2 + (y - 2)^2$
• Perpendicular distance $PM$ to the directrix $x - 9 = 0$: $PM = \frac{|x - 9|}{\sqrt{1^2 + 0^2}} = |x - 9|$

Step 3: Apply the fundamental conic equation ($SP^2 = e^2 \cdot PM^2$).
Substitute our expressions into the formula: $$ (x - 3)^2 + (y - 2)^2 = \left(\frac{1}{2}\right)^2 \cdot (x - 9)^2 $$ $$ (x - 3)^2 + (y - 2)^2 = \frac{1}{4} (x - 9)^2 $$
Step 4: Expand the binomial terms.
$$ (x^2 - 6x + 9) + (y^2 - 4y + 4) = \frac{1}{4} (x^2 - 18x + 81) $$ Multiply the entire equation by 4 to remove the fraction: $$ 4x^2 - 24x + 36 + 4y^2 - 16y + 16 = x^2 - 18x + 81 $$
Step 5: Group and simplify the terms to form the general equation.
Bring all terms to the left side: $$ (4x^2 - x^2) + 4y^2 + (-24x + 18x) - 16y + (52 - 81) = 0 $$ $$ 3x^2 + 4y^2 - 6x - 16y - 29 = 0 $$