If 4a2+9b2-c2+12ab=0, then the family of straight lines ax+by+c=0 is concurrent at
- (2,3)or(-2,-3)
- (-2,3)or(2,-3)
- (3,2)or(-3,2)
- (-3,2)or(2,3)
The Correct Option is A
Approach Solution - 1
\( 4a^2+9b^2+12ab-c^2=0\)
\(\Rightarrow (2a+3b)^2-c^2=0\)
\(\Rightarrow (2a+3b+c)(2a+3b-c)=0\)
\(\Rightarrow (2a+3b+c) =0 \,\,or\,\,(2a+3b-c)=0\)
\(\therefore ax+by+c=0 \) passes through the points (2,3) and (-2,-3)
Approach Solution -2
leading to \( (2a + 3b - c)(2a + 3b + c) = 0 \).
This gives us two cases: \( 2a + 3b = c \) or \( 2a + 3b = -c \).
Therefore, the equation of the line can be expressed as \( ax + by + c = 0 \) or \( ax + by + 2a + 3b = 0 \),
which simplifies to \( a(x + 2) + b(y + 3) = 0 \).
This equation holds true at point (-2, -3) or \( a(x - 2) + b(y - 3) = 0 \) at point (2, 3).
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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
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