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Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a
circle of radius r having AB as its diameter, then the slope of the line joining A and B can be -
circle of radius r having AB as its diameter, then the slope of the line joining A and B can be -
- a)-1/r
- b)1/r
- c)2/r
- d)3/r
Correct answer is option 'C'. Can you explain this answer?
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Let A and B be two distinct points on the parabola y2 = 4x. If the axi...
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Let A and B be two distinct points on the parabola y2 = 4x. If the axi...
Given information:
- The equation of the parabola is y^2 = 4x.
- A and B are two distinct points on the parabola.
- The axis of the parabola touches a circle with AB as its diameter.
To find:
The slope of the line joining A and B.
Solution:
Let's start by graphing the given parabola:
The equation of the parabola is y^2 = 4x.
We can rewrite this equation as y = ±√(4x).
We know that the parabola is symmetric with respect to the y-axis, and the vertex is at the origin (0, 0).
Now let's consider the circle with AB as its diameter. The midpoint of AB will be the center of the circle.
Step 1: Find the coordinates of A and B.
Let's assume A has coordinates (a, 2√a) and B has coordinates (b, 2√b), where a and b are positive real numbers.
Step 2: Find the midpoint of AB.
The midpoint of AB will have coordinates ((a+b)/2, (√a + √b)).
Step 3: Find the radius of the circle.
The radius of the circle is half the length of AB. Using the distance formula, we can find the length of AB as follows:
AB = √((b-a)^2 + (2√b - 2√a)^2)
= √((b-a)^2 + 4(b-a))
= √((b-a)(b-a+4))
Therefore, the radius of the circle is r = AB/2 = √((b-a)(b-a+4))/2.
Step 4: Find the slope of the line joining A and B.
The slope of the line joining A and B can be found using the formula:
m = (2√b - 2√a)/(b - a)
= 2(√b - √a)/(b - a)
Step 5: Simplify the expression.
To simplify the expression, we can multiply both the numerator and denominator by (√b + √a) to eliminate the square roots:
m = 2(√b - √a)/(b - a) * (√b + √a)/(√b + √a)
= 2(b - a)/(√b + √a)
Step 6: Substitute the radius of the circle.
Since the circle has AB as its diameter, the radius is r = √((b-a)(b-a+4))/2.
Substituting this value into the expression for the slope:
m = 2(b - a)/(√b + √a)
= 2(b - a)/(√b + √a) * (√((b-a)(b-a+4))/√((b-a)(b-a+4)))
= 2(b - a)√((b-a)(b-a+4))/((b-a)(b-a+4))
= 2√((b-a)(b-a+4))/(b-a+4
- The equation of the parabola is y^2 = 4x.
- A and B are two distinct points on the parabola.
- The axis of the parabola touches a circle with AB as its diameter.
To find:
The slope of the line joining A and B.
Solution:
Let's start by graphing the given parabola:
The equation of the parabola is y^2 = 4x.
We can rewrite this equation as y = ±√(4x).
We know that the parabola is symmetric with respect to the y-axis, and the vertex is at the origin (0, 0).
Now let's consider the circle with AB as its diameter. The midpoint of AB will be the center of the circle.
Step 1: Find the coordinates of A and B.
Let's assume A has coordinates (a, 2√a) and B has coordinates (b, 2√b), where a and b are positive real numbers.
Step 2: Find the midpoint of AB.
The midpoint of AB will have coordinates ((a+b)/2, (√a + √b)).
Step 3: Find the radius of the circle.
The radius of the circle is half the length of AB. Using the distance formula, we can find the length of AB as follows:
AB = √((b-a)^2 + (2√b - 2√a)^2)
= √((b-a)^2 + 4(b-a))
= √((b-a)(b-a+4))
Therefore, the radius of the circle is r = AB/2 = √((b-a)(b-a+4))/2.
Step 4: Find the slope of the line joining A and B.
The slope of the line joining A and B can be found using the formula:
m = (2√b - 2√a)/(b - a)
= 2(√b - √a)/(b - a)
Step 5: Simplify the expression.
To simplify the expression, we can multiply both the numerator and denominator by (√b + √a) to eliminate the square roots:
m = 2(√b - √a)/(b - a) * (√b + √a)/(√b + √a)
= 2(b - a)/(√b + √a)
Step 6: Substitute the radius of the circle.
Since the circle has AB as its diameter, the radius is r = √((b-a)(b-a+4))/2.
Substituting this value into the expression for the slope:
m = 2(b - a)/(√b + √a)
= 2(b - a)/(√b + √a) * (√((b-a)(b-a+4))/√((b-a)(b-a+4)))
= 2(b - a)√((b-a)(b-a+4))/((b-a)(b-a+4))
= 2√((b-a)(b-a+4))/(b-a+4
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Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. Can you explain this answer?
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Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. 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 Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. Can you explain this answer?.
Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. 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 Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches acircle of radius r having AB as its diameter, then the slope of the line joining A and B can be -a)-1/rb)1/rc)2/rd)3/rCorrect answer is option 'C'. Can you explain this answer?.
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