Question

If the chord joining the points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \) on the parabola \( y^2 = 12x \) subtends a right angle at the vertex of the parabola, then \( x_1x_2 - y_1y_2 \) is equal to

• A. 292
• B. 288
• C. 284
• D. 280

Hint:

For a parabola \( y^2 = 4ax \), if a chord subtends a right angle at the vertex, then the product of parameters satisfies \( t_1t_2 = -4 \).

Answer 1

The Correct Option is B

To solve this problem, we need to find the value of \(x_1x_2 - y_1y_2\) given that a chord joining two points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) on the parabola \(y^2 = 12x\) subtends a right angle at the vertex of the parabola.

The parabola given is \(y^2 = 12x\). The vertex of this parabola is at \( (0, 0) \).

According to the problem, the segment joining the points \(P_1\) and \(P_2\) subtends a right angle at the vertex. The condition for two lines to be perpendicular involves the slopes of the lines:

If two lines have slopes \(m_1\) and \(m_2\), and they are perpendicular, then

\(m_1 \times m_2 = -1\)

For the chord \(P_1P_2\), subtending a right angle at the vertex means that the product of slopes of the tangents through \(P_1\) and \(P_2\) should be \(-1\).

The slope of the tangent at any point on the parabola \(y^2 = 4ax\) is given by:

m = y/x

For our specific parabola \(y^2 = 12x\), the value of \(4a = 12\), which gives \(a = 3\).

The general point on the parabola in parametric form is \(((at^2), (2at))\), where \(t\) is the parameter.

Let's express \(x_1\) and \(y_1\) in parametric form: \((3t_1^2, 6t_1)\)

Also, express \(x_2\) and \(y_2\) in parametric form: \((3t_2^2, 6t_2)\)

The slopes of the tangents at these points are:

\(m_1 = \frac{6t_1}{3t_1^2} = \frac{2}{t_1}\)

\(m_2 = \frac{6t_2}{3t_2^2} = \frac{2}{t_2}\)

For the slopes to be perpendicular, \(m_1 \cdot m_2 = -1\), meaning:

\(\left( \frac{2}{t_1} \right) \left( \frac{2}{t_2} \right) = -1\)

\(\frac{4}{t_1t_2} = -1 \Rightarrow t_1t_2 = -4\)

Using these parametric identities, we find:

\(x_1x_2 = 3t_1^2 \times 3t_2^2 = 9t_1^2t_2^2\)

\(y_1y_2 = 6t_1 \times 6t_2 = 36t_1t_2\)

We need to calculate \(x_1x_2 - y_1y_2\):

\(x_1x_2 - y_1y_2 = 9t_1^2t_2^2 - 36t_1t_2\)

Substituting \(t_1t_2 = -4\), we get:

\(9(t_1t_2)^2 - 36(t_1t_2) = 9(-4)^2 - 36(-4)\)

\(= 9 \times 16 + 144\)

\(= 144 + 144\)

\(= 288\)

Therefore, the value of \(x_1x_2 - y_1y_2\) is (288).