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PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. ΔPQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.
The circumradius of trapezium PBCQ is
- a)6√5
- b)3√6
- c)2√10
- d)5√3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
PQ is the double ordinate of the parabola y2 = 4x which passes through...
let the circumcenter of trapezium be T(n, 0)
Then PT = PB
Then PT = PB
or n = 7
radius √40 = 2 √10
radius √40 = 2 √10
Most Upvoted Answer
PQ is the double ordinate of the parabola y2 = 4x which passes through...
To find the circumradius of trapezium PBCQ, we need to first understand the properties of the given parabola and the triangle PQA.
Properties of the Parabola:
- The equation of the parabola is given as y^2 = 4x, which is a standard form of a parabola.
- The vertex of the parabola is at the origin (0,0) and the focus S lies on the positive x-axis.
- The distance from the vertex to the focus is called the focal length, which is given by the equation F = a, where 'a' is the distance from the vertex to the directrix.
- The directrix is a line perpendicular to the x-axis and equidistant from the vertex.
- The point P lies on the parabola and is the double ordinate, meaning its y-coordinate is twice the x-coordinate.
- The point S lies on the parabola and is the focus.
Properties of Triangle PQA:
- Triangle PQA is an isosceles right-angled triangle, where A lies on the axis of the parabola.
- Let the coordinates of point A be (a, 2a), where 'a' is a positive real number.
- The slope of the line PA can be calculated as (2a-0)/(a-0) = 2.
- The equation of the line PA can be written as y = 2x.
Finding Points B and C:
- To find the coordinates of point B, we substitute y = 2x in the equation of the parabola: (2x)^2 = 4x, which simplifies to x = 0 or x = 2.
- When x = 0, y = 0, so point B is (0, 0).
- When x = 2, y = 4, so point C is (2, 4).
Finding Point Q:
- Since point Q lies on the parabola, we can substitute the coordinates of Q (x, y) in the equation of the parabola: y^2 = 4x.
- Since Q lies on the line PA (y = 2x), we can substitute y = 2x in the equation of the parabola: (2x)^2 = 4x.
- Simplifying the equation, we get x = 2 or x = 0.
- When x = 2, y = 4, so point Q is (2, 4).
- When x = 0, y = 0, so point Q is (0, 0).
Finding the Circumradius of Trapezium PBCQ:
- The circumradius of a trapezium is the radius of the circle passing through all four vertices.
- The vertices of trapezium PBCQ are P(0, 0), B(0, 0), C(2, 4), and Q(2, 4).
- The distance between any two points can be calculated using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- The distance between P and C is d1 = sqrt((2-0)^2 + (4-0)^2) = sqrt(20).
- The distance between B and C is d2
Properties of the Parabola:
- The equation of the parabola is given as y^2 = 4x, which is a standard form of a parabola.
- The vertex of the parabola is at the origin (0,0) and the focus S lies on the positive x-axis.
- The distance from the vertex to the focus is called the focal length, which is given by the equation F = a, where 'a' is the distance from the vertex to the directrix.
- The directrix is a line perpendicular to the x-axis and equidistant from the vertex.
- The point P lies on the parabola and is the double ordinate, meaning its y-coordinate is twice the x-coordinate.
- The point S lies on the parabola and is the focus.
Properties of Triangle PQA:
- Triangle PQA is an isosceles right-angled triangle, where A lies on the axis of the parabola.
- Let the coordinates of point A be (a, 2a), where 'a' is a positive real number.
- The slope of the line PA can be calculated as (2a-0)/(a-0) = 2.
- The equation of the line PA can be written as y = 2x.
Finding Points B and C:
- To find the coordinates of point B, we substitute y = 2x in the equation of the parabola: (2x)^2 = 4x, which simplifies to x = 0 or x = 2.
- When x = 0, y = 0, so point B is (0, 0).
- When x = 2, y = 4, so point C is (2, 4).
Finding Point Q:
- Since point Q lies on the parabola, we can substitute the coordinates of Q (x, y) in the equation of the parabola: y^2 = 4x.
- Since Q lies on the line PA (y = 2x), we can substitute y = 2x in the equation of the parabola: (2x)^2 = 4x.
- Simplifying the equation, we get x = 2 or x = 0.
- When x = 2, y = 4, so point Q is (2, 4).
- When x = 0, y = 0, so point Q is (0, 0).
Finding the Circumradius of Trapezium PBCQ:
- The circumradius of a trapezium is the radius of the circle passing through all four vertices.
- The vertices of trapezium PBCQ are P(0, 0), B(0, 0), C(2, 4), and Q(2, 4).
- The distance between any two points can be calculated using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- The distance between P and C is d1 = sqrt((2-0)^2 + (4-0)^2) = sqrt(20).
- The distance between B and C is d2
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PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. PQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.The circumradius of trapezium PBCQ isa)65b)36c)210d)53Correct answer is option 'C'. Can you explain this answer?
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PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. PQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.The circumradius of trapezium PBCQ isa)65b)36c)210d)53Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. PQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.The circumradius of trapezium PBCQ isa)65b)36c)210d)53Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. PQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.The circumradius of trapezium PBCQ isa)65b)36c)210d)53Correct answer is option 'C'. Can you explain this answer?.
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