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```python?code_reference&code_event_index=2
\n" ); document.write( "# Given information
\n" ); document.write( "h = 4 # x-coordinate of the vertex
\n" ); document.write( "k = 3 # y-coordinate of the vertex
\n" ); document.write( "latus_rectum_length = 4
\n" ); document.write( "condition = \"4p > 0\"\r
\n" ); document.write( "\n" ); document.write( "# The axis of symmetry is y = k (y = 3), so the parabola is horizontal,
\n" ); document.write( "# with the standard form: (y - k)^2 = 4p(x - h)\r
\n" ); document.write( "\n" ); document.write( "# The length of the latus rectum is |4p|.
\n" ); document.write( "# |4p| = 4.
\n" ); document.write( "# Since 4p > 0, we must have 4p = 4.
\n" ); document.write( "four_p = latus_rectum_length\r
\n" ); document.write( "\n" ); document.write( "# The equation is (y - k)^2 = 4p(x - h)
\n" ); document.write( "# Substitute the values
\n" ); document.write( "equation_lhs = f\"(y - {k})^2\"
\n" ); document.write( "equation_rhs = f\"{four_p}(x - {h})\"\r
\n" ); document.write( "\n" ); document.write( "print(f\"Vertex (h, k): ({h}, {k})\")
\n" ); document.write( "print(f\"Axis of symmetry: y = {k}. (Horizontal parabola, form (y - k)^2 = 4p(x - h))\")
\n" ); document.write( "print(f\"Length of latus rectum |4p|: {latus_rectum_length}\")
\n" ); document.write( "print(f\"Condition: {condition}\")
\n" ); document.write( "print(f\"Since |4p| = {latus_rectum_length} and 4p > 0, we have 4p = {four_p}\")
\n" ); document.write( "print(f\"The equation of the parabola is: {equation_lhs} = {equation_rhs}\")\r
\n" ); document.write( "\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "```text?code_stdout&code_event_index=2
\n" ); document.write( "Vertex (h, k): (4, 3)
\n" ); document.write( "Axis of symmetry: y = 3. (Horizontal parabola, form (y - k)^2 = 4p(x - h))
\n" ); document.write( "Length of latus rectum |4p|: 4
\n" ); document.write( "Condition: 4p > 0
\n" ); document.write( "Since |4p| = 4 and 4p > 0, we have 4p = 4
\n" ); document.write( "The equation of the parabola is: (y - 3)^2 = 4(x - 4)\r
\n" ); document.write( "\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "The equation of the parabola can be found using the given information:\r
\n" ); document.write( "\n" ); document.write( "1. **Identify the Standard Form:**\r
\n" ); document.write( "\n" ); document.write( " * The vertex is at $(h, k) = (4, 3)$.
\n" ); document.write( " * The axis of symmetry is the horizontal line $y = 3$.
\n" ); document.write( " * Since the axis of symmetry is horizontal ($y=k$), the parabola opens horizontally, and its standard form is:
\n" ); document.write( " $$(y - k)^2 = 4p(x - h)$$\r
\n" ); document.write( "\n" ); document.write( "2. **Substitute the Vertex Coordinates:**\r
\n" ); document.write( "\n" ); document.write( " * Substituting $h=4$ and $k=3$ into the standard form gives:
\n" ); document.write( " $$(y - 3)^2 = 4p(x - 4)$$\r
\n" ); document.write( "\n" ); document.write( "3. **Determine the Value of $4p$ (Latus Rectum):**\r
\n" ); document.write( "\n" ); document.write( " * The length of the latus rectum is $|4p|$.
\n" ); document.write( " * We are given that the length of the latus rectum is $4$, so $|4p| = 4$.
\n" ); document.write( " * This implies $4p = 4$ or $4p = -4$.
\n" ); document.write( " * The condition $4p > 0$ tells us that we must choose the positive value:
\n" ); document.write( " $$4p = 4$$\r
\n" ); document.write( "\n" ); document.write( "4. **Final Equation:**\r
\n" ); document.write( "\n" ); document.write( " * Substitute $4p = 4$ back into the equation from step 2:
\n" ); document.write( " $$(y - 3)^2 = 4(x - 4)$$\r
\n" ); document.write( "\n" ); document.write( "The equation of the parabola is $\mathbf{(y - 3)^2 = 4(x - 4)}$. \n" ); document.write( "