If ax + by + c = 0 is normal to xy = 1, then determine if a and b are less than, greater than, or equal to zero.
If ax + by + c = 0 is normal to xy = 1, then determine if a and b are less than, greater than, or equal to zero.
Solution and Explanation
To determine the relationship between a and b, we need to find the slope of the line ax + by + c = 0 and compare it to the slope of the line xy = 1.
The line ax + by + c = 0 can be rewritten as y = (-a/b)x - c/b, where the slope of this line is -a/b.
The line xy = 1 can be rewritten as y = 1/x, which can also be written as y = x^(-1). The slope of this line is -1.
For the line ax + by + c = 0 to be normal to xy = 1, the product of their slopes should be -1:
(-a/b) * (-1) = 1
Simplifying the equation:
a/b = 1
This tells us that a and b must have the same sign. If both a and b are positive or negative, the product will be positive, not -1. Therefore, a and b must be either both positive or both negative.
In conclusion, a and b are either both greater than zero or both less than zero.
Top Questions on Straight lines
- The vertices of a triangle are A(–1, 3), B(–2, 2) and C(3, –1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :
- JEE Main - 2024
- Mathematics
- Straight lines
- If the sum of squares of all real values of \( \alpha \), for which the lines \( 2x - y + 3 = 0 \), \( 6x + 3y + 1 = 0 \) and \( \alpha x + 2y - 2 = 0 \) do not form a triangle \( p \), then the greatest integer less than or equal to \( p \) is ....
- JEE Main - 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
- A rectangle ABCD has its side parallel to the line y=2x and vertices A,B,D are on y=1,x=1 and x=-1 respectively. The coordinate of C can be
- WBJEE - 2023
- Mathematics
- Straight lines
- Ki are possible values of K for which lines \(Kx + 2y + 2 = 0\), \(2x + Ky + 3 = 0\), \(3x + 3y + K = 0\) are concurrent, then \(∑k_i\) has value.
- MHT CET - 2023
- Mathematics
- Straight lines
Questions Asked in MHT CET exam
- 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 disease is primarily spread by female Anopheles mosquitoes?
- MHT CET - 2024
- HIV and AIDS
- Total genetic content of an organism is called
- MHT CET - 2024
- Non-Mendelian Genetics
- How many ATP molecules are needed as an initial investment in the glycolytic cycle (normal glycolysis)?
- MHT CET - 2024
- Glycolysis
- If \(A=\begin{bmatrix} 2a & -3b\\ 3 & 2\end{bmatrix}\) and \(\text{adj}A = AA^T\), then 2a+3b is?
- MHT CET - 2023
- Matrices
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