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A tangent to a parabola x2 = 4ay meets the hyperbola xy = c2 in two points P and Q. The locus of middle point of PQ is
- a)ellipse
- b)parabola
- c)hyperbola
- d)circle
Correct answer is option 'B'. Can you explain this answer?
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A tangent to a parabola x2 = 4ay meets the hyperbola xy = c2 in two po...
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A tangent to a parabola x2 = 4ay meets the hyperbola xy = c2 in two po...
Given:
- The equation of the parabola is x^2 = 4ay.
- The equation of the hyperbola is xy = c^2.
- The tangent to the parabola intersects the hyperbola at two points P and Q.
To Find:
The locus of the midpoint of PQ.
Solution:
Step 1: Determine the Points of Intersection
To find the points of intersection of the tangent and the hyperbola, we need to solve their equations simultaneously.
Substitute x^2 = 4ay in the equation xy = c^2:
(4ay)y = c^2
4ay^2 = c^2
y^2 = c^2/4a
Substitute y^2 = c^2/4a in the equation x^2 = 4ay:
x^2 = 4a(c^2/4a)
x^2 = c^2
So, the points of intersection are (c, c) and (-c, -c).
Step 2: Determine the Midpoint of PQ
Let the coordinates of P be (x1, y1) and the coordinates of Q be (x2, y2).
The midpoint of PQ is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Step 3: Find the Equation of the Locus
We know that the midpoint of PQ lies on the line joining P and Q. The equation of this line can be found using the two-point form:
(y - y1)/(y2 - y1) = (x - x1)/(x2 - x1)
Substituting the coordinates of P and Q, we have:
(y - c)/(-c - c) = (x - c)/(-c - c)
(y - c)/(-2c) = (x - c)/(-2c)
Simplifying, we get:
(y - c) = (x - c)
Therefore, the equation of the locus is y = x - c.
Step 4: Analyzing the Locus
The equation y = x - c represents a straight line with a slope of 1 and a y-intercept of -c. This is the equation of a straight line, which is a parabola.
Conclusion:
The locus of the midpoint of PQ is a parabola. Therefore, the correct answer is option B.
- The equation of the parabola is x^2 = 4ay.
- The equation of the hyperbola is xy = c^2.
- The tangent to the parabola intersects the hyperbola at two points P and Q.
To Find:
The locus of the midpoint of PQ.
Solution:
Step 1: Determine the Points of Intersection
To find the points of intersection of the tangent and the hyperbola, we need to solve their equations simultaneously.
Substitute x^2 = 4ay in the equation xy = c^2:
(4ay)y = c^2
4ay^2 = c^2
y^2 = c^2/4a
Substitute y^2 = c^2/4a in the equation x^2 = 4ay:
x^2 = 4a(c^2/4a)
x^2 = c^2
So, the points of intersection are (c, c) and (-c, -c).
Step 2: Determine the Midpoint of PQ
Let the coordinates of P be (x1, y1) and the coordinates of Q be (x2, y2).
The midpoint of PQ is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Step 3: Find the Equation of the Locus
We know that the midpoint of PQ lies on the line joining P and Q. The equation of this line can be found using the two-point form:
(y - y1)/(y2 - y1) = (x - x1)/(x2 - x1)
Substituting the coordinates of P and Q, we have:
(y - c)/(-c - c) = (x - c)/(-c - c)
(y - c)/(-2c) = (x - c)/(-2c)
Simplifying, we get:
(y - c) = (x - c)
Therefore, the equation of the locus is y = x - c.
Step 4: Analyzing the Locus
The equation y = x - c represents a straight line with a slope of 1 and a y-intercept of -c. This is the equation of a straight line, which is a parabola.
Conclusion:
The locus of the midpoint of PQ is a parabola. Therefore, the correct answer is option B.
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A tangent to a parabola x2 = 4ay meets the hyperbola xy = c2 in two points P and Q. The locus of middle point of PQ isa)ellipseb)parabolac)hyperbolad)circleCorrect answer is option 'B'. Can you explain this answer?
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