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The area of the triangle formed by the tangents from the point (4, 3) to the circle x2+y2=9 and the line joining their points of contact is
- a)192/25 sq. unit
- b)202/25 sq. unit
- c)16/25 sq. unit
- d)144/25 sq. unit
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The area of the triangle formed by the tangents from the point (4, 3)t...
From P(4, 3) two tangents PT and PT′ are drawn to the circle x2+y2=9 with O(0, 0) as centre and r=3. To find the area of ΔPTT′ .
Let R be the point of intersection of OP and TT′,
Then, we can prove by simple geometry that OP is perpendicular bisector of TT′.
The equation of chord of contact from a point (x1, y1) to a circle x2+y2=r2 is xx1+yy1=r2
Hence, the equation of chord of contact TT′ from the point (4, 3) to the circle x2+y2=9 is 4x+3y=9
The length of perpendicular from a point (x1, y1) on a line
Now, OR= length of the perpendicular from O to TT′ is 
Now, radius of circle OT=3
In triangle OTR, by Pythagoras theorem, we have
In triangle OTR, by Pythagoras theorem, we have
The distance between the points (x1, y1) and (x2, y2) is 
∴ Area of the triangle PTT′ will be twice area of ΔPRT
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The area of the triangle formed by the tangents from the point (4, 3)t...
To find the area of the triangle formed by the tangents from the point (4, 3) to the circle x² + y² = 9 and the line joining their points of contact, we can follow these steps:
1. Find the equation of the tangent line:
Let the equation of the tangent line be y = mx + c. Since the line passes through the point (4, 3), we can substitute these values into the equation to get:
3 = 4m + c
2. Find the coordinates of the points of contact:
Substituting y = mx + c into the equation of the circle, we get:
x² + (mx + c)² = 9
Simplifying this equation, we get:
(1 + m²) x² + 2mcx + c² - 9 = 0
This equation represents a quadratic equation in terms of x. The line is a tangent to the circle, so the discriminant of this quadratic equation should be zero:
(2mc)² - 4(1 + m²)(c² - 9) = 0
Simplifying this equation and rearranging, we get:
5m² + 12mc - 4c² - 36 = 0
This is a quadratic equation in terms of m. Solving this equation for m, we get two possible values for m.
3. Substitute the values of m into the equation of the tangent line:
For each value of m obtained in the previous step, substitute it into the equation y = mx + c to get the equations of the two tangent lines.
4. Find the points of contact using the equations of the tangent lines:
Substitute the equations of the tangent lines obtained in the previous step into the equation 3 = 4m + c to find the values of c. This will give us the coordinates of the two points of contact.
5. Find the area of the triangle:
Once we have the coordinates of the points of contact, we can use the formula for the area of a triangle given its three vertices to find the area of the triangle formed by the tangents and the line joining their points of contact.
By following these steps, we can find the area of the triangle formed by the tangents from the point (4, 3) to the circle x² + y² = 9 and the line joining their points of contact.
1. Find the equation of the tangent line:
Let the equation of the tangent line be y = mx + c. Since the line passes through the point (4, 3), we can substitute these values into the equation to get:
3 = 4m + c
2. Find the coordinates of the points of contact:
Substituting y = mx + c into the equation of the circle, we get:
x² + (mx + c)² = 9
Simplifying this equation, we get:
(1 + m²) x² + 2mcx + c² - 9 = 0
This equation represents a quadratic equation in terms of x. The line is a tangent to the circle, so the discriminant of this quadratic equation should be zero:
(2mc)² - 4(1 + m²)(c² - 9) = 0
Simplifying this equation and rearranging, we get:
5m² + 12mc - 4c² - 36 = 0
This is a quadratic equation in terms of m. Solving this equation for m, we get two possible values for m.
3. Substitute the values of m into the equation of the tangent line:
For each value of m obtained in the previous step, substitute it into the equation y = mx + c to get the equations of the two tangent lines.
4. Find the points of contact using the equations of the tangent lines:
Substitute the equations of the tangent lines obtained in the previous step into the equation 3 = 4m + c to find the values of c. This will give us the coordinates of the two points of contact.
5. Find the area of the triangle:
Once we have the coordinates of the points of contact, we can use the formula for the area of a triangle given its three vertices to find the area of the triangle formed by the tangents and the line joining their points of contact.
By following these steps, we can find the area of the triangle formed by the tangents from the point (4, 3) to the circle x² + y² = 9 and the line joining their points of contact.
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The area of the triangle formed by the tangents from the point (4, 3)to the circle x2+y2=9and the line joining their points of contact isa)192/25 sq. unitb)202/25 sq. unitc)16/25 sq. unitd)144/25 sq. unitCorrect answer is option 'A'. Can you explain this answer?
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The area of the triangle formed by the tangents from the point (4, 3)to the circle x2+y2=9and the line joining their points of contact isa)192/25 sq. unitb)202/25 sq. unitc)16/25 sq. unitd)144/25 sq. unitCorrect 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 The area of the triangle formed by the tangents from the point (4, 3)to the circle x2+y2=9and the line joining their points of contact isa)192/25 sq. unitb)202/25 sq. unitc)16/25 sq. unitd)144/25 sq. unitCorrect 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 The area of the triangle formed by the tangents from the point (4, 3)to the circle x2+y2=9and the line joining their points of contact isa)192/25 sq. unitb)202/25 sq. unitc)16/25 sq. unitd)144/25 sq. unitCorrect answer is option 'A'. Can you explain this answer?.
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