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P, Q are two points on the rectangular hyperbola (x – 1)(y – 2) = c2, O is the centre of hyperbola, also tangent at P is perpendicular to OQ and meets OQ at N such that (OQ)(ON) = 4
One of the equation of directrix of the hyperbola is
- a)x + y – 2 = 0
- b)x + y - 2√2 = 0
- c)x + y – 4 = 0
- d)x + y – 5 = 0
Correct answer is option 'A'. Can you explain this answer?
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P, Q are two points on the rectangular hyperbola (x 1)(y 2) = c2, O ...
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P, Q are two points on the rectangular hyperbola (x 1)(y 2) = c2, O ...
To find the equation of the directrix of the hyperbola, we need to consider the given conditions.
Let's start by finding the coordinates of the points P and Q on the hyperbola.
Given equation of the hyperbola is (x-1)(y-2) = c^2.
Let P = (x1, y1) be a point on the hyperbola. Substituting these values into the equation of the hyperbola, we get:
(x1-1)(y1-2) = c^2
Similarly, let Q = (x2, y2) be another point on the hyperbola. Substituting these values into the equation of the hyperbola, we get:
(x2-1)(y2-2) = c^2
Since the hyperbola is symmetric about the y-axis, the coordinates of P and Q will be of the form (a, b) and (-a, b) respectively.
So, we have:
(x1-1)(y1-2) = c^2
(x2-1)(y2-2) = c^2
Substituting a for x1 and b for y1, we get:
(a-1)(b-2) = c^2
Substituting -a for x2 and b for y2, we get:
(-a-1)(b-2) = c^2
Now, let's consider the tangent at P which is perpendicular to OQ.
The slope of the line passing through P and Q is given by:
m = (y2 - y1) / (x2 - x1)
Since the tangent at P is perpendicular to OQ, the product of their slopes will be -1.
So, the slope of the line OQ is given by:
m1 = -1 / m
Since the line OQ passes through the origin O(0,0), the equation of the line OQ is given by:
y = m1x
Substituting the value of m1, we get:
y = (-1 / m)(x)
Now, let's consider the point N where the tangent at P intersects OQ.
The coordinates of N can be found by solving the equations of the line OQ and the tangent at P.
Substituting the equation of the tangent at P into the equation of OQ, we get:
(-1 / m)(x) = (y - y1) / (x - x1)
Simplifying the equation, we get:
(-1 / m)(x - x1) = (y - y1)
Since the point N lies on the line OQ, its coordinates satisfy the equation of OQ.
Substituting the coordinates of N into the equation of OQ, we get:
(-1 / m)(xN) = (yN)
Now, let's consider the given condition that (OQ)(ON) = 4.
Substituting the equations of OQ and ON, we get:
(-1 / m)(xN) * (yN) = 4
Since m = (y2 - y1) / (x2 - x1), we can rewrite the equation as:
(-1 / ((y2 - y1) / (x2 - x1)))(xN)(yN) = 4
Let's start by finding the coordinates of the points P and Q on the hyperbola.
Given equation of the hyperbola is (x-1)(y-2) = c^2.
Let P = (x1, y1) be a point on the hyperbola. Substituting these values into the equation of the hyperbola, we get:
(x1-1)(y1-2) = c^2
Similarly, let Q = (x2, y2) be another point on the hyperbola. Substituting these values into the equation of the hyperbola, we get:
(x2-1)(y2-2) = c^2
Since the hyperbola is symmetric about the y-axis, the coordinates of P and Q will be of the form (a, b) and (-a, b) respectively.
So, we have:
(x1-1)(y1-2) = c^2
(x2-1)(y2-2) = c^2
Substituting a for x1 and b for y1, we get:
(a-1)(b-2) = c^2
Substituting -a for x2 and b for y2, we get:
(-a-1)(b-2) = c^2
Now, let's consider the tangent at P which is perpendicular to OQ.
The slope of the line passing through P and Q is given by:
m = (y2 - y1) / (x2 - x1)
Since the tangent at P is perpendicular to OQ, the product of their slopes will be -1.
So, the slope of the line OQ is given by:
m1 = -1 / m
Since the line OQ passes through the origin O(0,0), the equation of the line OQ is given by:
y = m1x
Substituting the value of m1, we get:
y = (-1 / m)(x)
Now, let's consider the point N where the tangent at P intersects OQ.
The coordinates of N can be found by solving the equations of the line OQ and the tangent at P.
Substituting the equation of the tangent at P into the equation of OQ, we get:
(-1 / m)(x) = (y - y1) / (x - x1)
Simplifying the equation, we get:
(-1 / m)(x - x1) = (y - y1)
Since the point N lies on the line OQ, its coordinates satisfy the equation of OQ.
Substituting the coordinates of N into the equation of OQ, we get:
(-1 / m)(xN) = (yN)
Now, let's consider the given condition that (OQ)(ON) = 4.
Substituting the equations of OQ and ON, we get:
(-1 / m)(xN) * (yN) = 4
Since m = (y2 - y1) / (x2 - x1), we can rewrite the equation as:
(-1 / ((y2 - y1) / (x2 - x1)))(xN)(yN) = 4
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P, Q are two points on the rectangular hyperbola (x 1)(y 2) = c2, O is the centre of hyperbola, also tangent at P is perpendicular to OQ and meets OQ at N such that (OQ)(ON) = 4 One of the equation of directrix of the hyperbola is a)x + y 2 = 0 b)x + y - 22 = 0 c)x + y 4 = 0 d)x + y 5 = 0 Correct answer is option 'A'. Can you explain this answer?
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