Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A variable straight line of slope 4 intersects the hyperbola \(x y=1\) at two points. The locus of the point which divides the line segment between these two points in the ratio \(1: 2\) is (A) \(16 x^{2}+10 x y+y^{2}=2\) (B) \(16 x^{2}-10 x y+y^{2}=2\) (C) \(16 x^{2}+10 x y+y^{2}=4\) (D) none of these

Step-by-Step Solution

Answer

The correct locus is given by option (A) \(16x^2 + 10xy + y^2 = 2\).

Step 1: Understand the Problem

We are given a line with slope 4 intersecting a hyperbola at two points and need to find the locus of a point dividing the segment between these points in a 1:2 ratio. The intersection occurs on the hyperbola \(xy = 1\). Our task is to find the locus equation of the dividing point.

Step 2: Determine Line Equation

The slope of the line is 4, so it can be expressed in the form \(y = 4x + c\). We need to solve for the points of intersection between this line and the hyperbola \(xy = 1\).

Step 3: Solve Intersection Points

Substitute \(y = 4x + c\) into \(xy = 1\):\[x(4x + c) = 1 \Rightarrow 4x^2 + cx - 1 = 0.\] Solve this quadratic equation for \(x\). The solutions \(x_1\) and \(x_2\) are the x-coordinates of the intersection points.

Step 4: Apply Section Formula

The section formula gives us the coordinates of the point dividing the segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\), by the coordinates \(\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)\). Here, \(m:n = 1:2\).

Step 5: Express Coordinates in Terms of Locus Variables

Let the coordinates dividing the segment between \((x_1, y_1)\) and \((x_2, y_2)\) be \((x, y)\). Substitute \(x\) and \(y\) using the section formula and simplify the expressions getting involved with \(c\) from the coefficient of the intersection line.

Step 6: Derive Locus Equation

Substitute expressions for \(x\) and \(y\) derived from the section formula in the locus equation, solving the simplified expressions related to \(c\). By eliminating \(c\), derive a general equation for the locus.

Step 7: Match Derived Equation with Options

After simplification, match the derived equation with the provided options to find the correct choice.

Key Concepts

Hyperbola

A hyperbola is a type of conic section formed by intersecting a double-cone with a plane in such a way that the plane does not touch the base of the cone. It consists of two distinct branches that are mirror images of each other. The standard equation of a hyperbola that opens horizontally is \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]and if it opens vertically, the equation is:\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]Where

• \((h, k)\) is the center of the hyperbola.
• \(a\) is the distance from the center to each vertex.
• \(b\) is related to the distance between the asymptotes.

In the context of the given problem, we deal with the hyperbola \(xy = 1\), which is a rectangular hyperbola. This means the axes are asymptotes and the hyperbola is symmetrical about the lines \(x = y\) and \(x = -y\). This shape plays a crucial role when analyzing points of intersection with any straight line.

Intersection of Line and Curve

The intersection of a line and a curve involves finding the points where they meet. For a line of the form \(y = mx + c\), intersecting a curve given by a function such as \(xy = 1\), involves substituting \(y\) from the line equation into the curve equation. This results in a single equation in terms of \(x\).In our exercise, the intersection points are found by solving:\[ x(4x + c) = 1 \Rightarrow 4x^2 + cx - 1 = 0 \]This is a quadratic equation in terms of \(x\). The solutions \(x_1\) and \(x_2\) are the x-coordinates of the intersection points of the line and the hyperbola. These intersection points are essential to finding the locus of another related point, which is what the original problem aims to address.

Section Formula

The section formula is a vital tool in coordinate geometry to determine the coordinates of a point that divides the line segment joining two points in a given ratio. If you have points \((x_1, y_1)\) and \((x_2, y_2)\), and you want to find the point \((x, y)\) that divides this line segment in the ratio \(m:n\), you use the formula:\[\begin{align*} x & = \frac{mx_2 + nx_1}{m+n} \ y & = \frac{my_2 + ny_1}{m+n} \end{align*}\]In our problem, we apply this formula to the points \((x_1, y_1)\) and \((x_2, y_2)\). With a given ratio of \(1:2\), the formula helps us determine the coordinates of the point dividing the segment joining the line with the hyperbola. This point's coordinates are essential to derive the equation for its locus.

Quadratic Equation

In the context of the problem, a quadratic equation arises when determining the intersection of a line and a hyperbola. The general form of a quadratic equation is \[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants. Solving this equation means finding values of \(x\) that satisfy it, which can be done using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the problem, we solve the quadratic equation resulting from substituting the line equation \(y = 4x + c\) into the hyperbola equation \(xy = 1\):\[4x^2 + cx - 1 = 0\]The solutions to this equation, \(x_1\) and \(x_2\), are crucial for identifying the points of intersection, helping establish the basis for finding the point's locus dividing the segment in the desired ratio.