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Let PSQ be the focal chord of the parabola, y2 = 8x. If the length of SP = 6 then, l(SQ) is equal to(where S is the focus)
- a)3
- b)4
- c)6
- d)None
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Let PSQ be the focal chord of the parabola, y2= 8x. If the length of S...
Since the semi latus rectum of a parabola is the harmonic mean between the segment of any focal chord of a parabola, therefore,SP,4,SQ are in H.P.
⇒4=2(SP.SQ)/(SP+SQ)
⇒4=2*6.SQ/(6+SQ)
⇒SQ=3
⇒4=2(SP.SQ)/(SP+SQ)
⇒4=2*6.SQ/(6+SQ)
⇒SQ=3
Most Upvoted Answer
Let PSQ be the focal chord of the parabola, y2= 8x. If the length of S...
To solve this problem, we need to use the properties of a parabola and the definition of a focal chord.
Given:
- The equation of the parabola is y^2 = 8x.
- PSQ is the focal chord of the parabola.
- The length of SP is 6 units.
1. Definition of a focal chord:
A focal chord of a parabola is a chord that passes through the focus of the parabola. In this case, the focus (S) is located at the point (2, 0) because the equation of the parabola is y^2 = 8x.
2. Finding the coordinates of P:
Since P lies on the parabola, we can substitute the x-coordinate of P into the equation to find the y-coordinate.
Let the x-coordinate of P be x_P.
Then, the y-coordinate of P is given by y_P^2 = 8x_P.
3. Finding the coordinates of Q:
Since Q also lies on the parabola, we can substitute the x-coordinate of Q into the equation to find the y-coordinate.
Let the x-coordinate of Q be x_Q.
Then, the y-coordinate of Q is given by y_Q^2 = 8x_Q.
4. Using the distance formula:
The length of SP is given as 6 units. Using the distance formula, we can find the distance between S and P.
The distance formula is given by: d = √((x2 - x1)^2 + (y2 - y1)^2)
5. Using the properties of a focal chord:
According to the properties of a focal chord, the perpendicular distance from the midpoint of the focal chord to the directrix is equal to half the length of the focal chord.
6. Finding the equation of the directrix:
The equation of the directrix for the given parabola is x = -2. This can be found by using the definition of a parabola and the focus-directrix property.
7. Finding the midpoint of PSQ:
Since the midpoint of PSQ lies on the directrix, we can find its coordinates by substituting the x-coordinate of the midpoint into the equation of the directrix.
8. Finding the length of SQ:
Using the distance formula, we can find the distance between S and Q.
9. Comparing the lengths of SP and SQ:
Since we know that the length of SP is 6 units, and we have found the length of SQ, we can compare the two lengths to determine the correct answer.
Therefore, the correct answer is option 'A' (3 units).
Given:
- The equation of the parabola is y^2 = 8x.
- PSQ is the focal chord of the parabola.
- The length of SP is 6 units.
1. Definition of a focal chord:
A focal chord of a parabola is a chord that passes through the focus of the parabola. In this case, the focus (S) is located at the point (2, 0) because the equation of the parabola is y^2 = 8x.
2. Finding the coordinates of P:
Since P lies on the parabola, we can substitute the x-coordinate of P into the equation to find the y-coordinate.
Let the x-coordinate of P be x_P.
Then, the y-coordinate of P is given by y_P^2 = 8x_P.
3. Finding the coordinates of Q:
Since Q also lies on the parabola, we can substitute the x-coordinate of Q into the equation to find the y-coordinate.
Let the x-coordinate of Q be x_Q.
Then, the y-coordinate of Q is given by y_Q^2 = 8x_Q.
4. Using the distance formula:
The length of SP is given as 6 units. Using the distance formula, we can find the distance between S and P.
The distance formula is given by: d = √((x2 - x1)^2 + (y2 - y1)^2)
5. Using the properties of a focal chord:
According to the properties of a focal chord, the perpendicular distance from the midpoint of the focal chord to the directrix is equal to half the length of the focal chord.
6. Finding the equation of the directrix:
The equation of the directrix for the given parabola is x = -2. This can be found by using the definition of a parabola and the focus-directrix property.
7. Finding the midpoint of PSQ:
Since the midpoint of PSQ lies on the directrix, we can find its coordinates by substituting the x-coordinate of the midpoint into the equation of the directrix.
8. Finding the length of SQ:
Using the distance formula, we can find the distance between S and Q.
9. Comparing the lengths of SP and SQ:
Since we know that the length of SP is 6 units, and we have found the length of SQ, we can compare the two lengths to determine the correct answer.
Therefore, the correct answer is option 'A' (3 units).
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Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer?.
Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer?.
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Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of Let PSQ be the focal chord of the parabola, y2= 8x. If the length of SP = 6 then,l(SQ) is equal to(where S is the focus)a)3b)4c)6d)NoneCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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