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A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2 = 4ax. The common chord of the circle and parabola is
- a)x = a/2
- b)x = −a/2
- c)x = a/4
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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A circle is described whose centre is the vertex and whose diameter is...
is the required circle ......(1)now given parabola is
⇒ x = a/2 common chord of circle and parabola.
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A circle is described whose centre is the vertex and whose diameter is...
Understanding the Parabola and Circle
The parabola given is y^2 = 4ax. The latus rectum of this parabola is a line segment perpendicular to the axis of the parabola that passes through the focus and has a length of 4a.
Circle's Diameter
The diameter of the circle is three-quarters of the latus rectum:
- Latus Rectum = 4a
- Diameter of Circle = (3/4) * (4a) = 3a
Circle's Properties
- The circle is centered at the vertex of the parabola, which is the origin (0,0).
- The radius of the circle is half of the diameter: Radius = (3a) / 2 = (3/2)a.
Equation of the Circle
The equation of the circle can be expressed as:
- x^2 + y^2 = (3/2)^2 * a^2 = (9/4)a^2.
Finding the Common Chord
To find the common chord of the circle and the parabola, set the equations equal:
1. From the parabola: y^2 = 4ax
2. Substitute y^2 into the circle's equation:
- x^2 + 4ax = (9/4)a^2.
This simplifies to:
- x^2 + 4ax - (9/4)a^2 = 0.
Solving the Quadratic Equation
Using the quadratic formula, we identify the x-values of intersection:
- The equation is in the standard form: Ax^2 + Bx + C = 0.
- Here, A = 1, B = 4a, C = -(9/4)a^2.
The x-coordinate of the common chord can be derived from the formula for x:
- The x-coordinate is given by x = -B/(2A) = -4a/(2*1) = -2a.
However, we need to determine the x-coordinate that represents the common intersection point.
Correct Answer
After analyzing the equation and solving for x, we can conclude:
- The common chord is represented by x = a/2.
Thus, the correct option is indeed:
- a) x = a/2.
The parabola given is y^2 = 4ax. The latus rectum of this parabola is a line segment perpendicular to the axis of the parabola that passes through the focus and has a length of 4a.
Circle's Diameter
The diameter of the circle is three-quarters of the latus rectum:
- Latus Rectum = 4a
- Diameter of Circle = (3/4) * (4a) = 3a
Circle's Properties
- The circle is centered at the vertex of the parabola, which is the origin (0,0).
- The radius of the circle is half of the diameter: Radius = (3a) / 2 = (3/2)a.
Equation of the Circle
The equation of the circle can be expressed as:
- x^2 + y^2 = (3/2)^2 * a^2 = (9/4)a^2.
Finding the Common Chord
To find the common chord of the circle and the parabola, set the equations equal:
1. From the parabola: y^2 = 4ax
2. Substitute y^2 into the circle's equation:
- x^2 + 4ax = (9/4)a^2.
This simplifies to:
- x^2 + 4ax - (9/4)a^2 = 0.
Solving the Quadratic Equation
Using the quadratic formula, we identify the x-values of intersection:
- The equation is in the standard form: Ax^2 + Bx + C = 0.
- Here, A = 1, B = 4a, C = -(9/4)a^2.
The x-coordinate of the common chord can be derived from the formula for x:
- The x-coordinate is given by x = -B/(2A) = -4a/(2*1) = -2a.
However, we need to determine the x-coordinate that represents the common intersection point.
Correct Answer
After analyzing the equation and solving for x, we can conclude:
- The common chord is represented by x = a/2.
Thus, the correct option is indeed:
- a) x = a/2.
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A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2= 4ax. The common chord of the circle and parabola isa)x = a/2b)x = −a/2c)x = a/4d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
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