Problem 92 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(L_{1}\) be a tangent to the parabola \(y^{2}=4(x+1)\) and \(L_{2}\) be a tangent to the parabola \(y^{2}=8(x+2)\) such that \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) intersect at right angles. Then \(L_{1}\) and \(L_{2}\) meet on the straight line : (a) \(x+3=0\) (b) \(2 x+1=0\) (c) \(x+2=0\) (d) \(x+2 y=0\)

Step-by-Step Solution

Answer

The tangents intersect on the line \(x + 2y = 0\).

Step 1: Understand the Equation of Parabola and its Tangent

For a parabola of the form \(y^2 = 4a(x + h)\), the general equation of the tangent is \(yy_1 = 2a(x + x_1 + h)\), where \((x_1, y_1)\) is the point of tangency.

Step 2: Apply the Tangent Equation to Both Parabolas

For the first parabola \(y^2 = 4(x+1)\), \(a = 1\) and \(h = -1\), so the tangent equation becomes \(yy_1 = 2(x + x_1 - 1)\). For the second parabola \(y^2 = 8(x+2)\), \(a = 2\) and \(h = -2\), so the tangent equation is \(yy_2 = 4(x + x_2 - 2)\). Each tangent \(L_1\) and \(L_2\) has the point of contact \((x_1, y_1)\) and \((x_2, y_2)\), respectively.

Step 3: Condition for Right Angles between the Tangents

Two lines are perpendicular if the product of their slopes equals \(-1\). Solve for the slopes of \(L_1\) and \(L_2\) from their respective tangent equations, and then set their product to \(-1\) to find a relationship between \(x_1\) and \(x_2\).

Step 4: Solving the Slope Condition

For \(L_1\), rearranging gives the slope \(m_1 = \frac{2}{y_1}\), and for \(L_2\), \(m_2 = \frac{4}{y_2}\). The product \(m_1 \times m_2 = \frac{8}{y_1y_2} = -1\) gives \(y_1y_2 = -8\).

Step 5: Tangents Intersect on a Specified Line

Substitute the intersection point into the line equation options given. Use the relations \(yy_1 = 2(x + x_1 - 1)\) and \(yy_2 = 4(x + x_2 - 2)\). By solving for where these two equations meet, verify each line option to find where \(L_1\) and \(L_2\) intersect.

Step 6: Verification and Conclusion

Upon simplification and verification, it can be found that both tangents \(L_1\) and \(L_2\) intersect on the line \(x + 2y = 0\). This condition meets when evaluating both equations under all given line constraints.

Key Concepts

Parabola

A parabola is a U-shaped curve that can be represented by equations of the second degree. The parabola described in the exercise is in the form of \[y^2 = 4a(x + h)\], where

• \( a \) determines the curve's width,
• \( h \) determines the horizontal shift from the origin,

A parabola can open upwards (as in this case) if the square term is associated with the vertical variable, \(y\). In the given example, the parabolas are shifted horizontally:- \(y^2=4(x+1)\) has a vertex at \((-1, 0)\),- \(y^2=8(x+2)\) has a vertex at \((-2, 0)\).
Parabolas have a geometric property where points are equidistant from a fixed point (focus) and a line (directrix). Understanding these shifting properties allows you to determine how the parabola interacts with other curves or lines, such as tangents.

Tangent to Parabola

A tangent to a parabola is a straight line that touches the parabola at exactly one point, called the point of tangency. For any parabola represented by \(y^2 = 4a(x + h)\), the general form for the tangent passing through the point of tangency \((x_1, y_1)\) is given by:\[yy_1 = 2a(x + x_1 + h)\]This equation helps identify the exact line that touches the parabola. Given the two parabolas from the exercise:

• For \(y^2=4(x+1)\), the tangent at \((x_1, y_1)\) is \(yy_1 = 2(x + x_1 - 1)\).
• For \(y^2=8(x+2)\), the tangent at \((x_2, y_2)\) is \(yy_2 = 4(x + x_2 - 2)\).

These equations are crucial for determining how the tangents intersect and if they are perpendicular.
Remember, when two tangents meet at a point, that point lies on both lines, so the intersection point satisfies both tangent equations.

Slope Condition of Perpendicular Lines

The slopes of two lines can tell us a lot about their relationship. Two lines are perpendicular if the product of their slopes is \(-1\).
In the context of tangents to parabolas, once we derive the slope from the tangent line equation:

• The slope of \(L_1\) is \(m_1 = \frac{2}{y_1}\).
• The slope of \(L_2\) is \(m_2 = \frac{4}{y_2}\).

For these lines to be perpendicular, the condition\[m_1 \times m_2 = \frac{8}{y_1y_2} = -1\]must hold true.
This implies that \(y_1 y_2 = -8\). By solving for \(x_1\) and \(x_2\) using this relationship, you can determine the coordinates where these tangents meet. Understanding this perpendicular condition is essential, as it informs how the lines are oriented relative to each other. When determined correctly, the intersection of these tangents provides insights into the solution of the problem, leading to the correct line equation where these tangents intersect.