The equation of the straight line which passes through the point (-5,4) and is such that the position of it between them is divided by the given point in the ratio 1:2: is_____.
The equation of the straight line which passes through the point (-5,4) and is such that the position of it between them is divided by the given point in the ratio 1:2: is_____.
5x+5y+160=0
8x+5y+160=0
8x+5y+90=0
3x+9y+60=0
The Correct Option is B
Solution and Explanation
We need to find the equation of a straight line that passes through the point \((-5, 4)\) and divides the segment between two points (not specified in the query) in the ratio 1:2. Since the two points are not given, we interpret the problem as finding the line passing through \((-5, 4)\) such that it divides the segment between two points \(A\) and \(B\) (to be determined) in the ratio 1:2 at \((-5, 4)\).
1. Interpret the Ratio Condition:
The point \((-5, 4)\) divides the segment between two points \(A\) and \(B\) in the ratio 1:2. Without \(A\) and \(B\), let’s assume the problem might mean the line’s intercepts on the axes, or we need to find a line where \((-5, 4)\) divides some segment. A common interpretation is that \((-5, 4)\) lies on the line, and the ratio might refer to a segment defined by intercepts or another point. Let’s proceed by assuming we need the line equation, and the ratio might apply to a segment defined later. First, let’s find a general line through \((-5, 4)\).
2. General Equation of the Line:
A line passing through the point \((-5, 4)\) can be written in the point-slope form:
\( y - y_1 = m (x - x_1) \)
Using \((-5, 4)\):
\( y - 4 = m (x - (-5)) \)
\( y - 4 = m (x + 5) \)
\( y = m x + 5m + 4 \)
This is the equation of the line with slope \(m\), which we need to determine using the ratio condition.
3. Find the Intercepts of the Line:
From the line equation \( y = m x + 5m + 4 \):
- X-intercept: Set \( y = 0 \):
\( 0 = m x + 5m + 4 \)
\( m x = -5m - 4 \)
\( x = \frac{-5m - 4}{m} \)
So, the x-intercept is \( \left( \frac{-5m - 4}{m}, 0 \right) \). If \( m = 0 \), the line is \( y = 4 \), which has no x-intercept, so assume \( m \neq 0 \).
- Y-intercept: Set \( x = 0 \):
\( y = m(0) + 5m + 4 = 5m + 4 \)
So, the y-intercept is \( (0, 5m + 4) \).
4. Apply the Ratio Condition:
Assume \((-5, 4)\) divides the segment between the x-intercept \( \left( \frac{-5m - 4}{m}, 0 \right) \) (point \( A \)) and the y-intercept \( (0, 5m + 4) \) (point \( B \)) in the ratio 1:2. Using the section formula, if a point divides the segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( k:1 \), its coordinates are:
\( \left( \frac{k x_2 + x_1}{k+1}, \frac{k y_2 + y_1}{k+1} \right) \)
Here, \( A = \left( \frac{-5m - 4}{m}, 0 \right) \), \( B = (0, 5m + 4) \), and the ratio is 1:2, so \( k = \frac{1}{2} \). The coordinates of the point dividing \( AB \) should match \((-5, 4)\):
- X-coordinate:
\( \frac{\frac{1}{2} \cdot 0 + \frac{-5m - 4}{m}}{\frac{1}{2} + 1} = -5 \)
\( \frac{\frac{-5m - 4}{m}}{\frac{3}{2}} = -5 \)
\( \frac{-5m - 4}{m} \cdot \frac{2}{3} = -5 \)
\( \frac{-5m - 4}{m} = -\frac{15}{2} \)
\( -5m - 4 = -\frac{15}{2} m \)
\( -5m + \frac{15}{2} m = 4 \)
\( \frac{-10m + 15m}{2} = 4 \)
\( \frac{5m}{2} = 4 \)
\( 5m = 8 \)
\( m = \frac{8}{5} \)
- Y-coordinate:
\( \frac{\frac{1}{2} (5m + 4) + 0}{\frac{1}{2} + 1} = 4 \)
\( \frac{\frac{1}{2} (5m + 4)}{\frac{3}{2}} = 4 \)
\( \frac{5m + 4}{3} = 4 \)
\( 5m + 4 = 12 \)
\( 5m = 8 \)
\( m = \frac{8}{5} \)
Both coordinates give the same \( m \), confirming consistency.
5. Equation of the Line:
Substitute \( m = \frac{8}{5} \) into the line equation:
\( y = \frac{8}{5} x + 5 \left( \frac{8}{5} \right) + 4 \)
\( y = \frac{8}{5} x + 8 + 4 \)
\( y = \frac{8}{5} x + 12 \)
Multiply through by 5 to clear the fraction:
\( 5y = 8x + 60 \)
\( 8x - 5y + 60 = 0 \)
6. Verify:
- The line passes through \((-5, 4)\):
\( 8(-5) - 5(4) + 60 = -40 - 20 + 60 = 0 \), which holds.
- Check intercepts:
X-intercept (\( y = 0 \)): \( 8x + 60 = 0 \), \( x = -\frac{60}{8} = -\frac{15}{2} \), so point \( A = \left( -\frac{15}{2}, 0 \right) \).
Y-intercept (\( x = 0 \)): \( -5y + 60 = 0 \), \( y = 12 \), so point \( B = (0, 12) \).
- Ratio check: \((-5, 4)\) divides \( \left( -\frac{15}{2}, 0 \right) \) to \( (0, 12) \) in 1:2:
X: \( \frac{\frac{1}{2} \cdot 0 + \left(-\frac{15}{2}\right)}{\frac{1}{2} + 1} = \frac{-\frac{15}{2}}{\frac{3}{2}} = -5 \), matches.
Y: \( \frac{\frac{1}{2} \cdot 12 + 0}{\frac{1}{2} + 1} = \frac{6}{\frac{3}{2}} = 4 \), matches.
Final Answer:
The equation of the straight line is:
\( 8x - 5y + 60 = 0 \)
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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