Question

If $(2,3)$ is the vertex and $(3,2)$ is the focus of a parabola, then its equation is

• A. $x^2+2xy+y^2-18x-2y+35=0$
• B. $2x^2+4xy+2y^2-9x-y+17=0$
• C. $x^2+2xy+y^2-18x-2y+17=0$
• D. $x^2+4xy+4y^2-18x+2y+9=0$

Hint:

Vertex-Focus Parabola Method.
To find the equation of a parabola from vertex and focus:

• Determine the axis from slope between vertex and focus.
• Find the directrix using midpoint and perpendicularity.
• Apply parabola definition: Distance to focus equals distance to directrix.

Answer 1

The Correct Option is C

Let vertex $V = (2,3)$ and focus $S = (3,2)$.
The axis of the parabola is the line joining $V$ and $S$. Slope: \[ m = \frac{2 - 3}{3 - 2} = -1 \Rightarrow x + y = 5 \] Let $Z$ be point on axis such that $V$ is midpoint of $SZ$ (point on directrix). Solve: \[ \frac{3 + x_Z}{2} = 2 \Rightarrow x_Z = 1, \quad \frac{2 + y_Z}{2} = 3 \Rightarrow y_Z = 4 \] So directrix passes through $Z = (1,4)$, perpendicular to axis with slope $1$: $x - y + 3 = 0$ Now use parabola definition: \[ \sqrt{(x - 3)^2 + (y - 2)^2} = \frac{|x - y + 3|}{\sqrt{2}} \] Squaring: \[ (x - 3)^2 + (y - 2)^2 = \frac{(x - y + 3)^2}{2} \] \[ 2(x^2 - 6x + 9 + y^2 - 4y + 4) = x^2 - 2xy + y^2 + 6x - 6y + 9 \] Simplifying: \[ x^2 + y^2 + 2xy - 18x - 2y + 17 = 0 \]