Parabola
Equation of Parabola
Grade 11
Question:
<p>Consider the following lines:</p><p>$$L_1: x - y - 1 = 0$$</p><p>$$L_2: x + y - 5 = 0$$</p><p>$$L_3: y - 4 = 0$$</p><p>Let $L_1$ is axis to a parabola, $L_2$ is tangent at the vertex to this parabola and $L_3$ is another tangent to this parabola at some point $P$.</p><p><strong>The given parabola is equal to which of the following parabola?</strong></p>
<p>(a) $y^2 = 16\sqrt{2}x$</p>
<p>(b) $x^2 = -4\sqrt{2}y$</p>
<p>(c) $y^2 = -\sqrt{2}x$</p>
<p>(d) $y^2 = 8\sqrt{2}x$</p>
Step-by-Step Solution
Key Concept: The axis of the parabola is given by $L_1$, vertex tangent by $L_2$, and another tangent by $L_3$. Use these to determine the vertex location and focal parameter.
<p>Using the conditions that $L_1$ is the axis, $L_2$ is tangent at vertex, and $L_3$ is another tangent, we find the vertex and parameter of the parabola. The vertex lies at the intersection of $L_1$ and $L_2$, and the axis direction is along $L_1$. Using the tangent condition at $L_3$ and the parabola properties, the equation is determined.</p><p>∴ Answer is (d): $y^2 = 8\sqrt{2}x$</p>
Correct Answer: D