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: \[ x^2 + y^2 + 2xy - 18x - 2y + 17 = 0 \]

Answer 2

Step 1: Identify vertex and focus
Given vertex \(V = (2,3)\) and focus \(F = (3,2)\).
The parabola is defined as the set of points equidistant from the focus and the directrix.

Step 2: Find the axis of the parabola
The axis passes through vertex and focus.
Slope of axis = \(\frac{2 - 3}{3 - 2} = \frac{-1}{1} = -1\).
So, axis has slope \(-1\).

Step 3: Find the directrix
The directrix is a line perpendicular to the axis and equidistant from the vertex on the opposite side of the focus.
Distance between vertex and focus = \(\sqrt{(3-2)^2 + (2-3)^2} = \sqrt{1 + 1} = \sqrt{2}\).
The directrix lies \(\sqrt{2}\) units away from the vertex along the line perpendicular to axis.

Step 4: Equation of axis line
Passing through vertex \((2,3)\), slope \(-1\):
\[ y - 3 = -1(x - 2) \implies y = -x + 5 \]

Step 5: Equation of directrix line
Directrix is perpendicular to axis, so slope of directrix = reciprocal of negative slope of axis = \(1\).
Directrix passes through point \(D\), which is \(\sqrt{2}\) units from vertex in direction opposite to focus along axis.
Direction vector from vertex to focus is \((3-2, 2-3) = (1, -1)\).
Unit vector in this direction: \(\frac{1}{\sqrt{2}}(1, -1)\).
Opposite direction vector = \(-\frac{1}{\sqrt{2}}(1, -1) = \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\).
Point on directrix:
\[ D = V + \sqrt{2} \times \text{opposite unit vector} = (2,3) + \sqrt{2} \times \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) = (2 - 1, 3 + 1) = (1,4) \]
Equation of directrix with slope 1 passing through (1,4):
\[ y - 4 = 1(x - 1) \implies y = x + 3 \]

Step 6: Write the parabola equation
For any point \((x,y)\), distance to focus equals distance to directrix:
\[ \sqrt{(x-3)^2 + (y-2)^2} = \frac{|y - x - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|y - x - 3|}{\sqrt{2}} \]
Square both sides:
\[ (x - 3)^2 + (y - 2)^2 = \frac{(y - x - 3)^2}{2} \]
Multiply both sides by 2:
\[ 2(x - 3)^2 + 2(y - 2)^2 = (y - x - 3)^2 \]

Step 7: Expand and simplify
Left side:
\[ 2[(x^2 - 6x + 9) + (y^2 - 4y + 4)] = 2(x^2 + y^2 - 6x - 4y + 13) = 2x^2 + 2y^2 - 12x - 8y + 26 \]
Right side:
\[ (y - x - 3)^2 = y^2 - 2xy - 6y + x^2 + 6x + 9 \]

Equate:
\[ 2x^2 + 2y^2 - 12x - 8y + 26 = x^2 + y^2 - 2xy - 6y + 6x + 9 \]
Bring all terms to left side:
\[ 2x^2 - x^2 + 2y^2 - y^2 - 12x - 6x - 8y + 6y + 26 - 9 + 2xy = 0 \]\[ x^2 + y^2 + 2xy - 18x - 2y + 17 = 0 \]

Conclusion:
The equation of the parabola is:
\[ x^2 + 2xy + y^2 - 18x - 2y + 17 = 0 \]