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On the parabola y2 = 8 x if one extremity of a focal chord is (1 ∕ 2, − 2) then its other extremity is
- a)(2, 2)
- b)(1 ∕ 8, − 8)
- c)(8, 1/8)
- d)(8, 8)
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, &...
We can start by finding the equation of the parabola y^2 = 8x. This is a standard form equation for a parabola that opens to the right with vertex at the origin. The focus is located at (2,0) and the directrix is the line x = -2.
Next, we can use the definition of a focal chord to find the other endpoint of the chord. A focal chord is a line segment that passes through the focus and has endpoints on the parabola. Since we know one endpoint is (1,2), we can find the slope of the line passing through that point and the focus:
m = (0 - 2) / (2 - 1) = -2
The equation of the line passing through (1,2) with slope -2 is:
y - 2 = -2(x - 1)
Simplifying, we get:
y = -2x + 4
Now we need to find where this line intersects the parabola. Substituting y = -2x + 4 into the equation of the parabola, we get:
(-2x + 4)^2 = 8x
Expanding and simplifying, we get:
4x^2 - 16x + 16 = 0
Dividing both sides by 4, we get:
x^2 - 4x + 4 = 0
This factors as:
(x - 2)^2 = 0
So the only solution is x = 2. Substituting this back into the equation of the line, we get:
y = -2(2) + 4 = 0
So the other endpoint of the focal chord is (2,0).
To find the length of the focal chord, we can use the distance formula:
d = sqrt((2 - 1)^2 + (0 - 2)^2) = sqrt(2)
Therefore, the length of the focal chord is sqrt(2) units.
Next, we can use the definition of a focal chord to find the other endpoint of the chord. A focal chord is a line segment that passes through the focus and has endpoints on the parabola. Since we know one endpoint is (1,2), we can find the slope of the line passing through that point and the focus:
m = (0 - 2) / (2 - 1) = -2
The equation of the line passing through (1,2) with slope -2 is:
y - 2 = -2(x - 1)
Simplifying, we get:
y = -2x + 4
Now we need to find where this line intersects the parabola. Substituting y = -2x + 4 into the equation of the parabola, we get:
(-2x + 4)^2 = 8x
Expanding and simplifying, we get:
4x^2 - 16x + 16 = 0
Dividing both sides by 4, we get:
x^2 - 4x + 4 = 0
This factors as:
(x - 2)^2 = 0
So the only solution is x = 2. Substituting this back into the equation of the line, we get:
y = -2(2) + 4 = 0
So the other endpoint of the focal chord is (2,0).
To find the length of the focal chord, we can use the distance formula:
d = sqrt((2 - 1)^2 + (0 - 2)^2) = sqrt(2)
Therefore, the length of the focal chord is sqrt(2) units.
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On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. 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 On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. Can you explain this answer?.
On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. 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 On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for On the parabola y2 = 8 x if one extremity of a focal chord is (1 2, − 2)then its other extremity isa)(2, 2)b)(1 8, − 8)c)(8, 1/8)d)(8, 8)Correct answer is option 'D'. Can you explain this answer?.
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