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If tangents are drawn to a circle x2 + y2 = 1 from each of the point on the line 2x + y = 4, then chord of contact passes through a fixed point whose coordinates are –
- a)(1/2, 1/4)
- b)(1/4, 1/2)
- c)(2,4)
- d)(4 ,2)
Correct answer is option 'A'. Can you explain this answer?
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If tangents are drawn to a circle x2 + y2 = 1 from each of the point o...
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If tangents are drawn to a circle x2 + y2 = 1 from each of the point o...
To find the fixed point, we need to find the equation of the tangents and then find the point of intersection of these tangents.
Given the equation of the circle: x^2 + y^2 = 1
And the equation of the line: 2x + y = 4
To find the equation of the tangents, we need to find the point of contact (x1, y1) for each tangent.
Substituting y = 4 - 2x into the equation of the circle, we get:
x^2 + (4 - 2x)^2 = 1
x^2 + 16 - 16x + 4x^2 = 1
5x^2 - 16x + 15 = 0
Solving this quadratic equation, we get:
(x - 1)(5x - 15) = 0
x = 1 or x = 3
For x = 1, substituting back into the equation of the line, we get:
2(1) + y = 4
y = 2
So the first point of contact (x1, y1) is (1, 2).
For x = 3, substituting back into the equation of the line, we get:
2(3) + y = 4
y = -2
So the second point of contact (x2, y2) is (3, -2).
Now, we need to find the equation of the line passing through these two points of contact.
The slope of the line passing through (1, 2) and (3, -2) is:
m = (y2 - y1) / (x2 - x1)
m = (-2 - 2) / (3 - 1)
m = -4 / 2
m = -2
Using the point-slope form of the line, we have:
y - y1 = m(x - x1)
y - 2 = -2(x - 1)
y - 2 = -2x + 2
y = -2x + 4
So the equation of the chord of contact is y = -2x + 4.
To find the fixed point, we need to find the intersection of this chord with the line 2x + y = 4.
Substituting y = -2x + 4 into the equation of the line, we get:
2x + (-2x + 4) = 4
2x - 2x + 4 = 4
4 = 4
This equation is always true, which means the chord of contact passes through all points on the line 2x + y = 4.
Therefore, the fixed point is the intersection point of the line 2x + y = 4 with itself, which is (2, 0).
Given the equation of the circle: x^2 + y^2 = 1
And the equation of the line: 2x + y = 4
To find the equation of the tangents, we need to find the point of contact (x1, y1) for each tangent.
Substituting y = 4 - 2x into the equation of the circle, we get:
x^2 + (4 - 2x)^2 = 1
x^2 + 16 - 16x + 4x^2 = 1
5x^2 - 16x + 15 = 0
Solving this quadratic equation, we get:
(x - 1)(5x - 15) = 0
x = 1 or x = 3
For x = 1, substituting back into the equation of the line, we get:
2(1) + y = 4
y = 2
So the first point of contact (x1, y1) is (1, 2).
For x = 3, substituting back into the equation of the line, we get:
2(3) + y = 4
y = -2
So the second point of contact (x2, y2) is (3, -2).
Now, we need to find the equation of the line passing through these two points of contact.
The slope of the line passing through (1, 2) and (3, -2) is:
m = (y2 - y1) / (x2 - x1)
m = (-2 - 2) / (3 - 1)
m = -4 / 2
m = -2
Using the point-slope form of the line, we have:
y - y1 = m(x - x1)
y - 2 = -2(x - 1)
y - 2 = -2x + 2
y = -2x + 4
So the equation of the chord of contact is y = -2x + 4.
To find the fixed point, we need to find the intersection of this chord with the line 2x + y = 4.
Substituting y = -2x + 4 into the equation of the line, we get:
2x + (-2x + 4) = 4
2x - 2x + 4 = 4
4 = 4
This equation is always true, which means the chord of contact passes through all points on the line 2x + y = 4.
Therefore, the fixed point is the intersection point of the line 2x + y = 4 with itself, which is (2, 0).
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If tangents are drawn to a circle x2 + y2 = 1 from each of the point on the line 2x + y = 4, then chord of contact passes through a fixed point whose coordinates are –a)(1/2, 1/4)b)(1/4, 1/2)c)(2,4)d)(4 ,2)Correct answer is option 'A'. Can you explain this answer?
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If tangents are drawn to a circle x2 + y2 = 1 from each of the point on the line 2x + y = 4, then chord of contact passes through a fixed point whose coordinates are –a)(1/2, 1/4)b)(1/4, 1/2)c)(2,4)d)(4 ,2)Correct answer is option 'A'. 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 tangents are drawn to a circle x2 + y2 = 1 from each of the point on the line 2x + y = 4, then chord of contact passes through a fixed point whose coordinates are –a)(1/2, 1/4)b)(1/4, 1/2)c)(2,4)d)(4 ,2)Correct answer is option 'A'. 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 tangents are drawn to a circle x2 + y2 = 1 from each of the point on the line 2x + y = 4, then chord of contact passes through a fixed point whose coordinates are –a)(1/2, 1/4)b)(1/4, 1/2)c)(2,4)d)(4 ,2)Correct answer is option 'A'. Can you explain this answer?.
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