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If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq ≠ 0) are bisected by the x –axis, then (1999 - 2 Marks)
- a)p2 = q2
- b)p2 = 8q2
- c)P2 < 8q2
- d)p2 > 8q2.
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If two distinct chords, drawn from the point (p, q) on the circle x2 +...
The given equation of the circle is x2 + y2 – px – qy = 0, pq ≠ 0 Let the chord drawn from (p, q) is bisected by x-axis at point (x1, 0).
Then equation of chord is
Then equation of chord is
(using T = S1)As it passes through (p, q), therefore,
As through (p,q) two distinct chords can be drawn.
∴ Roots of above equation be real and distinct, i.e., D > 0.
⇒ 9p2 – 4 × 2 (p2 + q2) > 0 ⇒ p2 > 8q2
Most Upvoted Answer
If two distinct chords, drawn from the point (p, q) on the circle x2 +...
We can use the equation of the circle to find the coordinates of the points where the chords intersect the circle. Let the two chords be AB and CD, and let their intersection point be E.
First, we find the equation of the line passing through A and B. Since A and B are both on the circle, they satisfy the equation x^2 + y^2 = px + qy. We can write the equation of the line passing through A and B in point-slope form:
(y - q)/(x - p) = (yB - yA)/(xB - xA)
where (xA, yA) and (xB, yB) are the coordinates of A and B respectively. Solving for y, we get:
y = [(xB - xA)/(yB - yA)](x - p) + q
Similarly, we can find the equation of the line passing through C and D:
y = [(xD - xC)/(yD - yC)](x - p) + q
To find the coordinates of E, we solve the system of equations:
x^2 + y^2 = px + qy (equation of the circle)
y = [(xB - xA)/(yB - yA)](x - p) + q (equation of line AB)
y = [(xD - xC)/(yD - yC)](x - p) + q (equation of line CD)
Substituting the equation of line AB into the equation of the circle, we get a quadratic equation in x:
x^2 + [(xB - xA)/(yB - yA)](x - p) + q)^2 = px + q[(xB - xA)/(yB - yA)](x - p) + q^2
Similarly, substituting the equation of line CD into the equation of the circle, we get another quadratic equation in x:
x^2 + [(xD - xC)/(yD - yC)](x - p) + q)^2 = px + q[(xD - xC)/(yD - yC)](x - p) + q^2
We can solve these two quadratic equations for x, and then substitute back into either of the line equations to find y. This gives us the coordinates of E.
Once we have the coordinates of E, we can find the lengths of the chords AB and CD using the distance formula:
AB = sqrt((xA - xB)^2 + (yA - yB)^2)
CD = sqrt((xC - xD)^2 + (yC - yD)^2)
We can then verify that AB and CD are distinct by checking that they are not equal.
First, we find the equation of the line passing through A and B. Since A and B are both on the circle, they satisfy the equation x^2 + y^2 = px + qy. We can write the equation of the line passing through A and B in point-slope form:
(y - q)/(x - p) = (yB - yA)/(xB - xA)
where (xA, yA) and (xB, yB) are the coordinates of A and B respectively. Solving for y, we get:
y = [(xB - xA)/(yB - yA)](x - p) + q
Similarly, we can find the equation of the line passing through C and D:
y = [(xD - xC)/(yD - yC)](x - p) + q
To find the coordinates of E, we solve the system of equations:
x^2 + y^2 = px + qy (equation of the circle)
y = [(xB - xA)/(yB - yA)](x - p) + q (equation of line AB)
y = [(xD - xC)/(yD - yC)](x - p) + q (equation of line CD)
Substituting the equation of line AB into the equation of the circle, we get a quadratic equation in x:
x^2 + [(xB - xA)/(yB - yA)](x - p) + q)^2 = px + q[(xB - xA)/(yB - yA)](x - p) + q^2
Similarly, substituting the equation of line CD into the equation of the circle, we get another quadratic equation in x:
x^2 + [(xD - xC)/(yD - yC)](x - p) + q)^2 = px + q[(xD - xC)/(yD - yC)](x - p) + q^2
We can solve these two quadratic equations for x, and then substitute back into either of the line equations to find y. This gives us the coordinates of E.
Once we have the coordinates of E, we can find the lengths of the chords AB and CD using the distance formula:
AB = sqrt((xA - xB)^2 + (yA - yB)^2)
CD = sqrt((xC - xD)^2 + (yC - yD)^2)
We can then verify that AB and CD are distinct by checking that they are not equal.
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If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq ≠ 0) are bisected by the x –axis, then (1999 - 2 Marks)a)p2 = q2b)p2 = 8q2c)P2 < 8q2d)p2 > 8q2.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 If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq ≠ 0) are bisected by the x –axis, then (1999 - 2 Marks)a)p2 = q2b)p2 = 8q2c)P2 < 8q2d)p2 > 8q2.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 If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq ≠ 0) are bisected by the x –axis, then (1999 - 2 Marks)a)p2 = q2b)p2 = 8q2c)P2 < 8q2d)p2 > 8q2.Correct answer is option 'D'. Can you explain this answer?.
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