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A straight line which make equal intercepts on +ve x and y axes and which is at a distance '1' unit from the origin intersects the straight line y = 2x + 3 + √2 at (x0, y0) then 2x0 + y0
- a)3 + √2
- b)√2 - 1
- c)1
- d)0
Correct answer is option 'B'. Can you explain this answer?
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A straight line which make equal intercepts on +ve x and y axes and wh...
Equation of the straight line having equal intercepts is x + y = k and proceed.
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A straight line which make equal intercepts on +ve x and y axes and wh...
To find the equation of the straight line that intersects both the x and y axes at equal intercepts and is at a distance of 1 unit from the origin, we can first find the equation of the line that passes through the origin and is perpendicular to the given line y = 2x - 3.
The slope of the given line y = 2x - 3 is 2. The slope of a line perpendicular to this line is the negative reciprocal of the slope, which is -1/2.
So, the equation of the line passing through the origin and perpendicular to y = 2x - 3 is y = (-1/2)x.
To find the point where this line is at a distance of 1 unit from the origin, we can use the distance formula:
1 = √((x - 0)^2 + (y - 0)^2)
1 = √(x^2 + y^2)
Squaring both sides of the equation:
1 = x^2 + y^2
Substituting y = (-1/2)x, we get:
1 = x^2 + (-1/2)x^2
1 = (3/2)x^2
2/3 = x^2
Taking the square root of both sides:
±√(2/3) = x
Since the line intersects the x-axis at equal intercepts, the x-coordinate of the point where it intersects the x-axis is ±√(2/3). Similarly, since the line intersects the y-axis at equal intercepts, the y-coordinate of the point where it intersects the y-axis is ±√(2/3).
Therefore, the four points where the line intersects the x and y axes are:
(√(2/3), 0)
(-√(2/3), 0)
(0, √(2/3))
(0, -√(2/3))
The equation of the line that intersects both the x and y axes at equal intercepts and is at a distance of 1 unit from the origin can be determined using two of these points. Let's use the points (√(2/3), 0) and (0, √(2/3)).
First, let's find the slope of the line passing through these two points:
slope = (y2 - y1) / (x2 - x1)
slope = (√(2/3) - 0) / (0 - √(2/3))
slope = (√(2/3)) / (-√(2/3))
slope = -1
Using the point-slope form of a line, we have:
y - y1 = m(x - x1)
y - √(2/3) = -1(x - 0)
y - √(2/3) = -x
Simplifying the equation, we get:
y = -x + √(2/3)
Therefore, the equation of the straight line that intersects both the x and y axes at equal intercepts and is at a distance of 1 unit from the origin intersects the line y = 2x - 3 at the point of intersection (-1/7, -1/2)
The slope of the given line y = 2x - 3 is 2. The slope of a line perpendicular to this line is the negative reciprocal of the slope, which is -1/2.
So, the equation of the line passing through the origin and perpendicular to y = 2x - 3 is y = (-1/2)x.
To find the point where this line is at a distance of 1 unit from the origin, we can use the distance formula:
1 = √((x - 0)^2 + (y - 0)^2)
1 = √(x^2 + y^2)
Squaring both sides of the equation:
1 = x^2 + y^2
Substituting y = (-1/2)x, we get:
1 = x^2 + (-1/2)x^2
1 = (3/2)x^2
2/3 = x^2
Taking the square root of both sides:
±√(2/3) = x
Since the line intersects the x-axis at equal intercepts, the x-coordinate of the point where it intersects the x-axis is ±√(2/3). Similarly, since the line intersects the y-axis at equal intercepts, the y-coordinate of the point where it intersects the y-axis is ±√(2/3).
Therefore, the four points where the line intersects the x and y axes are:
(√(2/3), 0)
(-√(2/3), 0)
(0, √(2/3))
(0, -√(2/3))
The equation of the line that intersects both the x and y axes at equal intercepts and is at a distance of 1 unit from the origin can be determined using two of these points. Let's use the points (√(2/3), 0) and (0, √(2/3)).
First, let's find the slope of the line passing through these two points:
slope = (y2 - y1) / (x2 - x1)
slope = (√(2/3) - 0) / (0 - √(2/3))
slope = (√(2/3)) / (-√(2/3))
slope = -1
Using the point-slope form of a line, we have:
y - y1 = m(x - x1)
y - √(2/3) = -1(x - 0)
y - √(2/3) = -x
Simplifying the equation, we get:
y = -x + √(2/3)
Therefore, the equation of the straight line that intersects both the x and y axes at equal intercepts and is at a distance of 1 unit from the origin intersects the line y = 2x - 3 at the point of intersection (-1/7, -1/2)
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