If the end points of chord of parabola \(y^2 = 12x\) are \((x_1, y_1)\) and \((x_2, y_2)\) and it subtend \(90^\circ\) at the vertex of parabola then \((x_1x_2 - y_1y_2)\) equals :
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For a parabola \(y^2=4ax\), if a chord joining points with parameters \(t_1\) and \(t_2\) subtends a right angle at the vertex, the condition is always \(t_1t_2 = -4\). This is a very useful property to remember.
Step 1: Understanding the Question:
We are given a parabola and a chord whose endpoints are \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). This chord subtends a right angle (90°) at the vertex of the parabola. We need to find the value of the expression \(x_1x_2 - y_1y_2\).
Step 2: Key Formula or Approach:
The equation of the parabola is \(y^2 = 12x\). Comparing this with the standard form \(y^2 = 4ax\), we get \(4a = 12\), so \(a=3\). The vertex of this parabola is at the origin, V(0,0).
Let the endpoints of the chord be represented in parametric form. For a parabola \(y^2=4ax\), any point can be written as \((at^2, 2at)\).
So, let \(P = (at_1^2, 2at_1)\) and \(Q = (at_2^2, 2at_2)\).
The slope of the line segment joining the vertex to P is \(m_1 = \frac{2at_1 - 0}{at_1^2 - 0} = \frac{2}{t_1}\).
The slope of the line segment joining the vertex to Q is \(m_2 = \frac{2at_2 - 0}{at_2^2 - 0} = \frac{2}{t_2}\).
Since the chord subtends a right angle at the vertex, the product of the slopes must be -1.
\[ m_1 m_2 = -1 \Rightarrow \left(\frac{2}{t_1}\right) \left(\frac{2}{t_2}\right) = -1 \Rightarrow \frac{4}{t_1t_2} = -1 \Rightarrow t_1t_2 = -4 \]
This is the condition for a chord to subtend a right angle at the vertex.
Step 3: Detailed Explanation:
Now we need to calculate \(x_1x_2 - y_1y_2\).
The coordinates are:
\(x_1 = at_1^2 = 3t_1^2\)
\(y_1 = 2at_1 = 6t_1\)
\(x_2 = at_2^2 = 3t_2^2\)
\(y_2 = 2at_2 = 6t_2\)
Let's compute the products:
\[ x_1x_2 = (3t_1^2)(3t_2^2) = 9(t_1t_2)^2 \]
Since \(t_1t_2 = -4\), we have:
\[ x_1x_2 = 9(-4)^2 = 9(16) = 144 \]
Now, for the y-coordinates:
\[ y_1y_2 = (6t_1)(6t_2) = 36(t_1t_2) \]
Since \(t_1t_2 = -4\), we have:
\[ y_1y_2 = 36(-4) = -144 \]
Step 4: Final Answer:
The expression we need to evaluate is \(x_1x_2 - y_1y_2\).
\[ x_1x_2 - y_1y_2 = 144 - (-144) = 144 + 144 = 288 \]
Thus, the value of the expression is 288. Alternatively, from the slope condition, \(\frac{y_1}{x_1}\frac{y_2}{x_2} = -1 \Rightarrow y_1y_2 = -x_1x_2\).
So, \(x_1x_2 - y_1y_2 = x_1x_2 - (-x_1x_2) = 2x_1x_2\).
Also, \(y_1^2y_2^2 = (4ax_1)(4ax_2) = 16a^2x_1x_2\). And \(y_1^2y_2^2 = (-x_1x_2)^2 = x_1^2x_2^2\).
Equating them gives \(16a^2x_1x_2 = x_1^2x_2^2 \Rightarrow x_1x_2 = 16a^2\).
The required value is \(2x_1x_2 = 2(16a^2) = 32a^2 = 32(3^2) = 32(9) = 288\).
