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A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate -3/5, then which one of the following options is correct?
- a)6 ≤ m < 8
- b)2 ≤ m < 4
- c)4 ≤ m < 6
- d)–3 ≤ m < –1
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25...
⇒ m2 – 5m + 6 = 0
⇒ m = 2, 3
Most Upvoted Answer
A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25...
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
Since the line intersects the circle at (x1, y1) and (x2, y2), we can substitute these values into the equation of the line to get two equations.
For the first point (x1, y1):
y1 = m*x1 + b
For the second point (x2, y2):
y2 = m*x2 + b
Now we can solve these two equations simultaneously to find the values of m and b.
Subtracting the first equation from the second equation, we get:
y2 - y1 = m*x2 - m*x1
Simplifying further, we have:
y2 - y1 = m*(x2 - x1)
Dividing both sides by (x2 - x1), we get:
m = (y2 - y1) / (x2 - x1)
Now we can substitute this value of m back into one of the original equations to solve for b.
Using the first equation (y1 = m*x1 + b), we have:
b = y1 - m*x1
Now we have the values of m and b, and we can substitute them into the equation of the line to get the final equation.
Therefore, the equation of the line y = mx + b that intersects the circle (x - h)^2 + (y - k)^2 = r^2 at points (x1, y1) and (x2, y2) is:
y = ((y2 - y1) / (x2 - x1)) * x + (y1 - ((y2 - y1) / (x2 - x1)) * x1)
Since the line intersects the circle at (x1, y1) and (x2, y2), we can substitute these values into the equation of the line to get two equations.
For the first point (x1, y1):
y1 = m*x1 + b
For the second point (x2, y2):
y2 = m*x2 + b
Now we can solve these two equations simultaneously to find the values of m and b.
Subtracting the first equation from the second equation, we get:
y2 - y1 = m*x2 - m*x1
Simplifying further, we have:
y2 - y1 = m*(x2 - x1)
Dividing both sides by (x2 - x1), we get:
m = (y2 - y1) / (x2 - x1)
Now we can substitute this value of m back into one of the original equations to solve for b.
Using the first equation (y1 = m*x1 + b), we have:
b = y1 - m*x1
Now we have the values of m and b, and we can substitute them into the equation of the line to get the final equation.
Therefore, the equation of the line y = mx + b that intersects the circle (x - h)^2 + (y - k)^2 = r^2 at points (x1, y1) and (x2, y2) is:
y = ((y2 - y1) / (x2 - x1)) * x + (y1 - ((y2 - y1) / (x2 - x1)) * x1)
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A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. Can you explain this answer?
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A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. 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 A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. Can you explain this answer?.
A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. 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 A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpointof the line segment PQ has x-coordinate -3/5,then which one of the following options is correct?a)6 ≤ m < 8b)2 ≤ m < 4c)4 ≤ m < 6d)–3 ≤ m < –1Correct answer is option 'B'. Can you explain this answer?.
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