If a pair of line given by (xcosα + ysinα)2 = (x2 + y2)sin2α are perpendicular. What is the value of α?
If a pair of line given by (xcosα + ysinα)2 = (x2 + y2)sin2α are perpendicular. What is the value of α?
Solution and Explanation
Step 1: Let the given equation be:
(xcosα + ysinα)2 = (x2 + y2)sin2α
Step 2: Expand the left side:
x2cos2α + 2xy sinα cosα + y2sin2α = x2sin2α + y2sin2α
x2cos2α + 2xy sinα cosα = x2sin2α
x2cos2α - x2sin2α + 2xy sinα cosα = 0
x2(cos2α - sin2α) + 2xy sinα cosα = 0
x2cos(2α) + xy sin(2α) = 0
Step 3: Recognize that this is a homogeneous equation of degree 2, representing a pair of lines passing through the origin.
Let the lines be y = m1x and y = m2x.
Step 4: Substitute y = mx into the equation:
x2cos(2α) + x(mx)sin(2α) = 0
x2(cos(2α) + m sin(2α)) = 0
Since x2 ≠ 0 in general,
cos(2α) + m sin(2α) = 0
m sin(2α) = -cos(2α)
m = -cos(2α) / sin(2α) = -cot(2α)
Step 5: Recognize that m1 and m2 are the roots of the equation cos(2α) + m sin(2α) = 0.
The sum of the roots is m1 + m2 = 0.
The product of the roots is m1m2 = cos(2α) / sin(2α).
Step 6: Apply the condition for perpendicular lines, m1m2 = -1.
So,
cos(2α) / sin(2α) = -1
cos(2α) = -sin(2α)
tan(2α) = -1
Step 7: Solve for 2α and then α:
2α = 3π/4 + nπ, n ∈ ℤ
α = 3π/8 + nπ/2
Step 8: Find the smallest positive value of α.
If n = 0, α = 3π/8.
If n = 1, α = 3π/8 + π/2 = 7π/8.
If n = 2, α = 3π/8 + π = 11π/8.
If n = 3, α = 3π/8 + 3π/2 = 15π/8.
The smallest positive value of α is 3π/8.
Final Answer: The final answer is 3π/8.
Top Questions on Straight lines
- The maximum area of a right-angled triangle with hypotenuse \( h \) is: (a) \( \frac{h^2}{2\sqrt{2}} \)
- VITEEE - 2024
- Mathematics
- Straight lines
- If lines $\frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2}$ and $\frac{x - 1}{3k} = \frac{y - 5}{1} = \frac{z - 6}{-5}$ are mutually perpendicular, then $k$ is equal to:
- KCET - 2024
- Mathematics
- Straight lines
- Which of the following cannot be the direction ratios of the straight line \(\frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1}\)?
- CUET (UG) - 2024
- Mathematics
- Straight lines
- The length of x-intercept made by the pair of lines \( 2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0 \) is:
- AP EAMCET - 2024
- Mathematics
- Straight lines
- If the slope of one of the pair of lines represented by \( 2x^2 + 3xy + Ky^2 = 0 \) is 2, then the angle between the pair of lines is:
- AP EAMCET - 2024
- Mathematics
- Straight lines
Questions Asked in MHT CET exam
- What is the oxidation state of Fe in \( \mathrm{Fe_3O_4} \)?
- MHT CET - 2024
- States of matter
- The number of ways of distributing \( 500 \) dissimilar boxes equally among \( 50 \) persons is
- MHT CET - 2024
- Permutations
- A committee of 11 members is to be formed out of 8 males and 5 females. If \( m \) is the number of ways the committee is formed with at least 6 males and \( n \) is the number of ways with at least 3 females, then:
- MHT CET - 2024
- Combinations
- Two monkeys off mass 10 kg and 8 kg are moving along a vertical light rope the former climbing up with an acceleration of 2 m/second square while the latter coming down with a uniform velocity of 2 m/sec square find the tension in the rope at the fixed support
- MHT CET - 2024
- tension
- Which of the following is ferromagnetic:
- MHT CET - 2024
- Magnetism and matter
Concepts Used:
Straight lines
A straight line is a line having the shortest distance between two points.
A straight line can be represented as an equation in various forms, as show in the image below:
The following are the many forms of the equation of the line that are presented in straight line-
1. Slope – Point Form
Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.
y – y0 = m (x – x0)
2. Two – Point Form
Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2) are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes
The slope of P2P = The slope of P1P2 , i.e.
\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)
Hence, the equation becomes:
y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)
3. Slope-Intercept Form
Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by
y – c =m( x - 0 )
As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if
y = m x +c