JEE Exam > JEE Questions > PASSAGE-1ABCD is a square of side length 2 un...
Start Learning for Free
PASSAGE-1
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Q. If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)
- a)ellipse
- b)hyper bola
- c)parabola
- d)pair of straight line
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle tou...
Let C' be the said circle touching C1 and L, so that C1 and C' are on the same side of L. Let us draw a line T parallel to L at a distance equal to the radius of circle C1, on opposite side of L.
Then the centre of C' is equidistant from the centre of C1 and from line T.
⇒ locus of centre of C' is a parabola.
Then the centre of C' is equidistant from the centre of C1 and from line T.
⇒ locus of centre of C' is a parabola.
Most Upvoted Answer
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle tou...
0)
Let O be the center of square ABCD and O1 be the center of circle C1. It is known that O1 is the midpoint of any side of the square. Let O2 be the center of circle C2. It is known that O2 is the intersection of the diagonals of the square, which is also the intersection of the perpendicular bisectors of any two sides of the square.
Let P be the center of the circle that touches the line L and circle C1 externally. Let the point of tangency between circle C1 and P be X. Let the point of tangency between line L and P be Y. Let the intersection of line PO2 and line L be Z.
Since circle C1 is tangent to all four sides of the square, O1X is perpendicular to AB and BC. Therefore, O1X is parallel to CD and DA. This means that triangle O1PX is isosceles, with O1P = O1X. Similarly, since line L is tangent to circle P, PY is perpendicular to line L. Therefore, triangle OPY is also isosceles, with OP = OY.
Since circle P is externally tangent to circle C1, O1P + PX = O1X. Substituting O1P = O1X/2, we get PX = O1X/2. Similarly, since circle P is externally tangent to line L, OP - PY = OY. Substituting OP = OY, we get PY = OY/2.
Now, consider triangle OPZ. Since line L is perpendicular to PO2, PY is also perpendicular to PO2. Therefore, angle OPZ is a right angle. Since O2Z is the perpendicular bisector of AB, triangle O2AB is isosceles, with O2A = O2B. Therefore, angle OAB = angle OBA. Since ABCD is a square, angle OAB = angle ABC = 90 degrees. Therefore, angle O2ZP = 90 - OAB = 0 degrees. This means that O2, Z, and P are collinear.
Since O2Z is the perpendicular bisector of AB, AZ = BZ = 2/2 = 1 unit. Therefore, OZ = sqrt(2^2 - 1^2) = sqrt(3) units. Since P is the center of a circle that touches both line L and circle C1, it must lie on the angle bisector of angle XO1Y. Therefore, angle PZO1 = angle PZC1. Since C1 and C2 are congruent circles with centers O1 and O2, respectively, angle PZC1 = angle PZO2. Therefore, angle PZO1 = angle PZO2.
Let M be the midpoint of PO2. Since triangle O2AB is isosceles, with O2A = O2B, M lies on the perpendicular bisector of AB. Therefore, AM = MB = sqrt(2) units. Since OZ = sqrt(3) units, OM = sqrt(2^2 + 3^2) = sqrt(13) units. Therefore, PM = PO2 - OM = 2 - sqrt(13) units.
Let N be the midpoint of O1X. Since triangle O1AB is isosceles
Let O be the center of square ABCD and O1 be the center of circle C1. It is known that O1 is the midpoint of any side of the square. Let O2 be the center of circle C2. It is known that O2 is the intersection of the diagonals of the square, which is also the intersection of the perpendicular bisectors of any two sides of the square.
Let P be the center of the circle that touches the line L and circle C1 externally. Let the point of tangency between circle C1 and P be X. Let the point of tangency between line L and P be Y. Let the intersection of line PO2 and line L be Z.
Since circle C1 is tangent to all four sides of the square, O1X is perpendicular to AB and BC. Therefore, O1X is parallel to CD and DA. This means that triangle O1PX is isosceles, with O1P = O1X. Similarly, since line L is tangent to circle P, PY is perpendicular to line L. Therefore, triangle OPY is also isosceles, with OP = OY.
Since circle P is externally tangent to circle C1, O1P + PX = O1X. Substituting O1P = O1X/2, we get PX = O1X/2. Similarly, since circle P is externally tangent to line L, OP - PY = OY. Substituting OP = OY, we get PY = OY/2.
Now, consider triangle OPZ. Since line L is perpendicular to PO2, PY is also perpendicular to PO2. Therefore, angle OPZ is a right angle. Since O2Z is the perpendicular bisector of AB, triangle O2AB is isosceles, with O2A = O2B. Therefore, angle OAB = angle OBA. Since ABCD is a square, angle OAB = angle ABC = 90 degrees. Therefore, angle O2ZP = 90 - OAB = 0 degrees. This means that O2, Z, and P are collinear.
Since O2Z is the perpendicular bisector of AB, AZ = BZ = 2/2 = 1 unit. Therefore, OZ = sqrt(2^2 - 1^2) = sqrt(3) units. Since P is the center of a circle that touches both line L and circle C1, it must lie on the angle bisector of angle XO1Y. Therefore, angle PZO1 = angle PZC1. Since C1 and C2 are congruent circles with centers O1 and O2, respectively, angle PZC1 = angle PZO2. Therefore, angle PZO1 = angle PZO2.
Let M be the midpoint of PO2. Since triangle O2AB is isosceles, with O2A = O2B, M lies on the perpendicular bisector of AB. Therefore, AM = MB = sqrt(2) units. Since OZ = sqrt(3) units, OM = sqrt(2^2 + 3^2) = sqrt(13) units. Therefore, PM = PO2 - OM = 2 - sqrt(13) units.
Let N be the midpoint of O1X. Since triangle O1AB is isosceles
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer?
Question Description
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. 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 PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer?.
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. 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 PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer?.
Solutions for PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice PASSAGE-1ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.Q.If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)a)ellipseb)hyper bolac)parabolad)pair of straight lineCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup