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If two distinct chords of a parabola y2 = 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :
Correct answer is '6'. Can you explain this answer?
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If two distinct chords of a parabola y2= 4ax passing through (a, 2a) a...
Any point on x + y = 1 can be taken as (t, 1–t)
The equation of chord with this as mid-point is y(1–t) –2a (x + t) = (1 – t)2 – 4at
It passes through (a, 2a)
So, t2 – 2t + 2a2 – 2a + 1 = 0
This should have two distinct real roots. So D > 0
⇒ a2 – a < 0
⇒ 0 < a < 1
So, length of latus rectum < 4 and 0 < a < 1
⇒ LR = 1, 2, 3
The equation of chord with this as mid-point is y(1–t) –2a (x + t) = (1 – t)2 – 4at
It passes through (a, 2a)
So, t2 – 2t + 2a2 – 2a + 1 = 0
This should have two distinct real roots. So D > 0
⇒ a2 – a < 0
⇒ 0 < a < 1
So, length of latus rectum < 4 and 0 < a < 1
⇒ LR = 1, 2, 3
Most Upvoted Answer
If two distinct chords of a parabola y2= 4ax passing through (a, 2a) a...
Problem: Find the sum of integral values of the length of possible latus rectums of the parabola y2= 4ax passing through (a, 2a) and whose two distinct chords are bisected on the line x + y = 1.
Solution:
Step 1: Find the equation of the parabola y2= 4ax passing through (a, 2a).
Since (a, 2a) lies on the parabola y2= 4ax, we can substitute a for x and 2a for y to get:
(2a)2 = 4a(a)
4a2 = 4a2
This is a true statement, so (a, 2a) lies on the parabola y2= 4ax.
Step 2: Find the equation of the line x + y = 1 in terms of y.
x + y = 1
y = 1 - x
Step 3: Find the equation of the line passing through (a, 2a) and bisecting the chord of the parabola y2= 4ax.
Since the chord is bisected on the line x + y = 1, we know that the midpoint of the chord lies on this line. Let (h, k) be the coordinates of the midpoint of the chord. Then:
h + k = 1 (equation of the line x + y = 1)
The midpoint of the chord also lies on the line passing through (a, 2a) and the midpoint of the chord. Let the equation of this line be y = mx + c. Then:
2a = ma + c (since (a, 2a) lies on the line)
k = 1/2(a + h) (since the chord is bisected at (h, k))
Substituting k = 1/2(a + h) into h + k = 1 gives:
h + 1/2(a + h) = 1
3h/2 + 1/2a = 1
3h + a = 2a
3h = a
h = a/3
Substituting h = a/3 into k = 1/2(a + h) gives:
k = 1/2(a + a/3)
k = 2a/3
So the line passing through (a, 2a) and bisecting the chord is y = mx + c, where:
m = (2a - 2a/3)/(a - a/3) = 2
c = 2a - 2am = -2a
The equation of this line is y = 2x - 2a.
Step 4: Find the equations of the two chords of the parabola that pass through (a, 2a).
Since (a, 2a) lies on the parabola y2= 4ax, we can substitute a for x and 2a for y to get:
(2a)2 = 4a(x)
x = a/2
So the point (a/2, 2a) lies on the parabola
Solution:
Step 1: Find the equation of the parabola y2= 4ax passing through (a, 2a).
Since (a, 2a) lies on the parabola y2= 4ax, we can substitute a for x and 2a for y to get:
(2a)2 = 4a(a)
4a2 = 4a2
This is a true statement, so (a, 2a) lies on the parabola y2= 4ax.
Step 2: Find the equation of the line x + y = 1 in terms of y.
x + y = 1
y = 1 - x
Step 3: Find the equation of the line passing through (a, 2a) and bisecting the chord of the parabola y2= 4ax.
Since the chord is bisected on the line x + y = 1, we know that the midpoint of the chord lies on this line. Let (h, k) be the coordinates of the midpoint of the chord. Then:
h + k = 1 (equation of the line x + y = 1)
The midpoint of the chord also lies on the line passing through (a, 2a) and the midpoint of the chord. Let the equation of this line be y = mx + c. Then:
2a = ma + c (since (a, 2a) lies on the line)
k = 1/2(a + h) (since the chord is bisected at (h, k))
Substituting k = 1/2(a + h) into h + k = 1 gives:
h + 1/2(a + h) = 1
3h/2 + 1/2a = 1
3h + a = 2a
3h = a
h = a/3
Substituting h = a/3 into k = 1/2(a + h) gives:
k = 1/2(a + a/3)
k = 2a/3
So the line passing through (a, 2a) and bisecting the chord is y = mx + c, where:
m = (2a - 2a/3)/(a - a/3) = 2
c = 2a - 2am = -2a
The equation of this line is y = 2x - 2a.
Step 4: Find the equations of the two chords of the parabola that pass through (a, 2a).
Since (a, 2a) lies on the parabola y2= 4ax, we can substitute a for x and 2a for y to get:
(2a)2 = 4a(x)
x = a/2
So the point (a/2, 2a) lies on the parabola
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If two distinct chords of a parabola y2= 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :Correct answer is '6'. Can you explain this answer?
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If two distinct chords of a parabola y2= 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :Correct answer is '6'. 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 If two distinct chords of a parabola y2= 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :Correct answer is '6'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two distinct chords of a parabola y2= 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :Correct answer is '6'. Can you explain this answer?.
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