Question Statement

An astronomer is studying the behaviour of light rays passing through a converging lens. The lens focuses the rays at the origin. The equation of rays is given by $3 x^{2} - 4 x y + y^{2} = 0$ .

Find:

The path of the individual light rays
The angle between the rays

Background and Explanation

The given equation is a homogeneous equation of second degree in $x$ and $y$ , which geometrically represents a pair of straight lines passing through the origin. By factorizing this quadratic expression into two linear factors, we obtain the equations of the individual light rays. Once we have the separate line equations, we can determine their slopes and calculate the angle between them using the tangent formula.

Solution

Finding the Individual Light Rays

We begin with the given homogeneous equation:

$3 x^{2} - 4 x y + y^{2} = 0$

To factorize this expression, we split the middle term $- 4 x y$ into $- 3 x y$ and $- x y$ (since $3 \times 1 = 3$ and we need factors that add to $- 4$ ):

$3 x^{2} - 3 x y - x y + y^{2} = 0$

Now we factor by grouping. Take $3 x$ common from the first two terms and $- y$ common from the last two terms:

$3 x (x - y) - y (x - y) = 0$

Notice that $(x - y)$ is a common factor in both terms. Factoring this out:

$(x - y) (3 x - y) = 0$

This product equals zero when either factor equals zero. Therefore, we obtain two separate linear equations representing the individual light rays:

$x - y = 0 \Rightarrow y = x$

and

$3 x - y = 0 \Rightarrow y = 3 x$

Thus, the paths of the individual light rays are the straight lines $y = x$ and $y = 3 x$ , both passing through the origin (the focal point of the lens).

Finding the Angle Between the Rays

To find the angle between these two lines, we first identify their slopes by comparing with the standard form $y = m x + c$ .

From the first ray $y = x$ : $Slope m_{1} = 1$

From the second ray $y = 3 x$ : $Slope m_{2} = 3$

The acute angle $θ$ between two lines with slopes $m_{1}$ and $m_{2}$ is given by the formula:

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

Substituting the values $m_{1} = 1$ and $m_{2} = 3$ :

$tan θ = \frac{3 - 1}{1 + ( 1 ) ( 3 )} = \frac{2}{1 + 3} = \frac{2}{4} = \frac{1}{2}$

Therefore:

$θ = tan^{- 1} (\frac{1}{2})$

Calculating the numerical value:

$θ \approx 26.5 6^{\circ}$

Key Formulas or Methods Used

Factorization by grouping: Splitting the middle term of a quadratic expression $a x^{2} + b x y + c y^{2}$ to factor it into two linear terms
Slope identification: For a line $y = m x$ , the coefficient $m$ represents the slope
Angle between two lines: $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ where $m_{1}$ and $m_{2}$ are the slopes of the two lines

Summary of Steps

Factorize the homogeneous equation $3 x^{2} - 4 x y + y^{2} = 0$ by splitting the middle term and grouping to get $(x - y) (3 x - y) = 0$ .
Identify the individual ray equations: $y = x$ (first ray) and $y = 3 x$ (second ray).
Determine slopes: Extract $m_{1} = 1$ and $m_{2} = 3$ from the line equations.
Apply angle formula: Substitute into $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ to get $tan θ = \frac{1}{2}$ .
Calculate: Find $θ = tan^{- 1} (0.5) \approx 26.5 6^{\circ}$ .

Question Statement

An astronomer is studying the behaviour of light rays passing through a converging lens. The lens focuses the rays at the origin. The equation of rays is given by $3 x^{2} - 4 x y + y^{2} = 0$ .

Find:

The path of the individual light rays
The angle between the rays

Background and Explanation

Solution

Finding the Individual Light Rays

We begin with the given homogeneous equation:

$3 x^{2} - 4 x y + y^{2} = 0$

To factorize this expression, we split the middle term $- 4 x y$ into $- 3 x y$ and $- x y$ (since $3 \times 1 = 3$ and we need factors that add to $- 4$ ):

$3 x^{2} - 3 x y - x y + y^{2} = 0$

Now we factor by grouping. Take $3 x$ common from the first two terms and $- y$ common from the last two terms:

$3 x (x - y) - y (x - y) = 0$

Notice that $(x - y)$ is a common factor in both terms. Factoring this out:

$(x - y) (3 x - y) = 0$

This product equals zero when either factor equals zero. Therefore, we obtain two separate linear equations representing the individual light rays:

$x - y = 0 \Rightarrow y = x$

and

$3 x - y = 0 \Rightarrow y = 3 x$

Thus, the paths of the individual light rays are the straight lines $y = x$ and $y = 3 x$ , both passing through the origin (the focal point of the lens).

Finding the Angle Between the Rays

To find the angle between these two lines, we first identify their slopes by comparing with the standard form $y = m x + c$ .

From the first ray $y = x$ : $Slope m_{1} = 1$

From the second ray $y = 3 x$ : $Slope m_{2} = 3$

The acute angle $θ$ between two lines with slopes $m_{1}$ and $m_{2}$ is given by the formula:

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

Substituting the values $m_{1} = 1$ and $m_{2} = 3$ :

$tan θ = \frac{3 - 1}{1 + ( 1 ) ( 3 )} = \frac{2}{1 + 3} = \frac{2}{4} = \frac{1}{2}$

Therefore:

$θ = tan^{- 1} (\frac{1}{2})$

Calculating the numerical value:

$θ \approx 26.5 6^{\circ}$

Key Formulas or Methods Used

Factorization by grouping: Splitting the middle term of a quadratic expression $a x^{2} + b x y + c y^{2}$ to factor it into two linear terms
Slope identification: For a line $y = m x$ , the coefficient $m$ represents the slope
Angle between two lines: $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ where $m_{1}$ and $m_{2}$ are the slopes of the two lines

Summary of Steps

Factorize the homogeneous equation $3 x^{2} - 4 x y + y^{2} = 0$ by splitting the middle term and grouping to get $(x - y) (3 x - y) = 0$ .
Identify the individual ray equations: $y = x$ (first ray) and $y = 3 x$ (second ray).
Determine slopes: Extract $m_{1} = 1$ and $m_{2} = 3$ from the line equations.
Apply angle formula: Substitute into $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ to get $tan θ = \frac{1}{2}$ .
Calculate: Find $θ = tan^{- 1} (0.5) \approx 26.5 6^{\circ}$ .

6.3 Q-8

Question Statement

Background and Explanation

Solution

Finding the Individual Light Rays

Finding the Angle Between the Rays

Key Formulas or Methods Used

Summary of Steps

6.3 Q-8

Question Statement

Background and Explanation

Solution

Finding the Individual Light Rays

Finding the Angle Between the Rays

Key Formulas or Methods Used

Summary of Steps