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Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)?
Most Upvoted Answer
Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the...
**Given Information:**
- A circle of radius 3
- A point P outside angle APB= 2 pi / 3 (120 degrees) the circle such that
- A and B are points of tangency from P to the circle.
- Line joining P and center of the circle cut the circle at C and D
- C is near to point P
**To Find:**
- Which statement(s) is/are correct?
- |Area of triangle ABC - Area of triangle ABD| = (3sqrt(3))/2
- 2 Area of triangle ABC - Area of triangle 9sqrt(3)
- Circum radius of triangle PAB/sqrt(3)
- Circum radius of triangle PAB is 2sqrt(3)
**Solution:**
Let O be the center of the given circle.
**Claim 1:**
- PA = PB = PC = PD = 3 (Given that P is outside the circle of radius 3)
- PO = 3 (Radius of the circle)
**Proof 1:**
- Let's draw the perpendiculars from O to the lines PA and PB. Let M be the point of intersection of the perpendicular from O to PA and N be the point of intersection of the perpendicular from O to PB.
- Since AM and BN are tangents to the circle, we have AM = AN = 3 (Radius of the circle)
- Triangle OMA and ONB are right triangles with hypotenuse equal to 3 and one of the sides equal to the radius of the circle. Therefore, OM = ON = sqrt(9 - 3^2) = sqrt(6)
- OP = sqrt(OM^2 + PM^2) = sqrt(6^2 + 3^2) = 3sqrt(3)
- Since OP is the perpendicular bisector of AB, we have PA = PB = 3
- Similarly, PC = PD = 3
**Claim 2:**
- Angle AOB = 120 degrees
- Triangle PAB is an equilateral triangle
**Proof 2:**
- Angle APB = 2 pi / 3 (Given)
- Angle AOB is the angle subtended by the arc AB. Therefore, AOB = 2APB = 4 pi / 3
- Since the sum of the angles of the triangle AOB is pi radians, we have Angle AOB = 120 degrees
- Since PA = PB, the perpendicular bisector of AB passes through O. Therefore, triangle PAB is an isosceles triangle.
- Since Angle APB = 120 degrees, Angle PAB = Angle PBA = (180 - 120)/2 = 30 degrees
- Therefore, Triangle PAB is an equilateral triangle
**Claim 3:**
- Area of triangle ABC = 9sqrt(3)/2
- Area of triangle ABD = 3sqrt(3)/2
**Proof 3:**
- Triangle PAB is an equilateral triangle with side length 3
- Height of the equilateral triangle is sqrt(3)/2 times the side length. Therefore, height of PAB = 3sqrt(3)/2
- Since AB is a chord of the circle and PC is the perpendicular bisector of AB, we have PC is
- A circle of radius 3
- A point P outside angle APB= 2 pi / 3 (120 degrees) the circle such that
- A and B are points of tangency from P to the circle.
- Line joining P and center of the circle cut the circle at C and D
- C is near to point P
**To Find:**
- Which statement(s) is/are correct?
- |Area of triangle ABC - Area of triangle ABD| = (3sqrt(3))/2
- 2 Area of triangle ABC - Area of triangle 9sqrt(3)
- Circum radius of triangle PAB/sqrt(3)
- Circum radius of triangle PAB is 2sqrt(3)
**Solution:**
Let O be the center of the given circle.
**Claim 1:**
- PA = PB = PC = PD = 3 (Given that P is outside the circle of radius 3)
- PO = 3 (Radius of the circle)
**Proof 1:**
- Let's draw the perpendiculars from O to the lines PA and PB. Let M be the point of intersection of the perpendicular from O to PA and N be the point of intersection of the perpendicular from O to PB.
- Since AM and BN are tangents to the circle, we have AM = AN = 3 (Radius of the circle)
- Triangle OMA and ONB are right triangles with hypotenuse equal to 3 and one of the sides equal to the radius of the circle. Therefore, OM = ON = sqrt(9 - 3^2) = sqrt(6)
- OP = sqrt(OM^2 + PM^2) = sqrt(6^2 + 3^2) = 3sqrt(3)
- Since OP is the perpendicular bisector of AB, we have PA = PB = 3
- Similarly, PC = PD = 3
**Claim 2:**
- Angle AOB = 120 degrees
- Triangle PAB is an equilateral triangle
**Proof 2:**
- Angle APB = 2 pi / 3 (Given)
- Angle AOB is the angle subtended by the arc AB. Therefore, AOB = 2APB = 4 pi / 3
- Since the sum of the angles of the triangle AOB is pi radians, we have Angle AOB = 120 degrees
- Since PA = PB, the perpendicular bisector of AB passes through O. Therefore, triangle PAB is an isosceles triangle.
- Since Angle APB = 120 degrees, Angle PAB = Angle PBA = (180 - 120)/2 = 30 degrees
- Therefore, Triangle PAB is an equilateral triangle
**Claim 3:**
- Area of triangle ABC = 9sqrt(3)/2
- Area of triangle ABD = 3sqrt(3)/2
**Proof 3:**
- Triangle PAB is an equilateral triangle with side length 3
- Height of the equilateral triangle is sqrt(3)/2 times the side length. Therefore, height of PAB = 3sqrt(3)/2
- Since AB is a chord of the circle and PC is the perpendicular bisector of AB, we have PC is
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Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)?
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Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)?.
Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)?.
Solutions for Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? in English & in Hindi are available as part of our courses for JEE.
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ample number of questions to practice Given a circle of radius 3 and a point P outside angle APB= 2 pi 3 the circle such that , where A and B are points of tangency from P to the circle. Line joining P and center of the circle cut the circle at C and D (Where C is near to point P), which statement(s) is/are correct. of the following |Area of triangle ABC - Area of triangle 9sqrt(3) is 2 Area of triangle ABC - Arca of triangle ABD| (3sqrt(3))/2 is 2 Circum radius of triangle PAB/s sqrt 3 Circum radius of triangle PAB is 2sqrt(3)? tests, examples and also practice JEE tests.
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