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The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ?
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The condition that the straight line joining the origin to the point o...
Problem: Determine the condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle ((x-3)^2)+((y-4)^2)=25.
Solution:
We can solve the problem by finding the coordinates of the point of intersection of the line and circle, and then finding the slope of the line joining the origin to that point.
Finding the coordinates of the point of intersection:
We will use substitution method to solve the system of equations consisting of the line and the circle.
Substituting y = (24 - 4x)/3 in the equation of the circle, we get:
((x-3)^2) + (((24 - 4x)/3) - 4)^2 = 25
Simplifying the above equation, we get:
25x^2 - 200x + 475 = 0
Solving the above quadratic equation, we get:
x = 3 and x = 19/5
Substituting these values of x in the equation of the line, we get:
When x = 3, y = 4
When x = 19/5, y = -2/5
Therefore, the two points of intersection are (3, 4) and (19/5, -2/5).
Finding the slope of the line joining the origin to the point of intersection:
The slope of the line joining the origin (0, 0) and the point (3, 4) is given by:
m1 = (4 - 0)/(3 - 0) = 4/3
The slope of the line joining the origin (0, 0) and the point (19/5, -2/5) is given by:
m2 = (-2/5 - 0)/(19/5 - 0) = -2/19
Condition for the two lines:
Since the slope of the line joining the origin to the point (3, 4) is positive and the slope of the line joining the origin to the point (19/5, -2/5) is negative, the two lines make equal angles with the x-axis.
Therefore, the correct option is (c) make equal angles with x-axis.
Final answer: The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle ((x-3)^2)+((y-4)^2)=25 is that the lines make equal angles with the x-axis.
Solution:
We can solve the problem by finding the coordinates of the point of intersection of the line and circle, and then finding the slope of the line joining the origin to that point.
Finding the coordinates of the point of intersection:
We will use substitution method to solve the system of equations consisting of the line and the circle.
Substituting y = (24 - 4x)/3 in the equation of the circle, we get:
((x-3)^2) + (((24 - 4x)/3) - 4)^2 = 25
Simplifying the above equation, we get:
25x^2 - 200x + 475 = 0
Solving the above quadratic equation, we get:
x = 3 and x = 19/5
Substituting these values of x in the equation of the line, we get:
When x = 3, y = 4
When x = 19/5, y = -2/5
Therefore, the two points of intersection are (3, 4) and (19/5, -2/5).
Finding the slope of the line joining the origin to the point of intersection:
The slope of the line joining the origin (0, 0) and the point (3, 4) is given by:
m1 = (4 - 0)/(3 - 0) = 4/3
The slope of the line joining the origin (0, 0) and the point (19/5, -2/5) is given by:
m2 = (-2/5 - 0)/(19/5 - 0) = -2/19
Condition for the two lines:
Since the slope of the line joining the origin to the point (3, 4) is positive and the slope of the line joining the origin to the point (19/5, -2/5) is negative, the two lines make equal angles with the x-axis.
Therefore, the correct option is (c) make equal angles with x-axis.
Final answer: The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle ((x-3)^2)+((y-4)^2)=25 is that the lines make equal angles with the x-axis.
Community Answer
The condition that the straight line joining the origin to the point o...
B) perpendicular. ...just solve circle and line. .U'll get 2 pts - A - ( 6,0 ) & B-( 0,8 ) ..one pt lie on X axis and other on y axis..so angle b/w OA and OB is 90 degree
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The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ?
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The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ?.
The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The condition that the straight line joining the origin to the point of intersection of the line 4x+3y=24 with the circle( (x-3)^2)+((y-4)^2)=25 are :a) coincident, b) perpendicular c) make equal angle with x axis d) none of above ?.
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