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PQ is a normal chord of the parabola y2 = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR is
- a)Equal to the length of the latus rectum
- b)Equal to the focal distance of the point P.
- c)Equal to twice the focal distance of the point P.
- d)Equal to the distance of the point P from the directrix
Correct answer is option 'C'. Can you explain this answer?
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PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex ...
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PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex ...
To solve this problem, let's analyze the given information step by step:
1. Equation of the Parabola: The given equation y^2 = 4ax represents a parabola with vertex A at the origin (0,0) and the focus at point F(a,0). The axis of symmetry is the x-axis.
2. Chord PQ: The line segment PQ is a chord of the parabola passing through a point P(x1, y1). Since PQ is a normal chord, it is perpendicular to the axis of symmetry.
3. Line parallel to AQ: Through point P, a line is drawn parallel to AQ (the axis of symmetry). Let's call the point of intersection of this line with the x-axis as R(x2, 0).
We need to find the length of AR.
4. Finding the equation of line PQ: Since PQ is perpendicular to the axis of symmetry, its slope is the negative inverse of the slope of AQ. The slope of AQ can be found using the derivative of the parabola equation:
dy/dx = 2a/x
At point P(x1, y1), the slope of the tangent line to the parabola is dy/dx = 2a/x1. Therefore, the slope of PQ is -x1/(2a).
Using the slope-intercept form of a line, the equation of line PQ is:
y - y1 = (-x1/(2a))(x - x1) ... (1)
5. Finding the equation of line PR: Since PR is parallel to AQ, its slope is the same as that of AQ, which is 0 (since AQ is parallel to the x-axis).
Using the point-slope form of a line, the equation of line PR is:
y - 0 = 0(x - x2)
y = 0 ... (2)
6. Finding the point of intersection R: To find the x-coordinate of point R, we substitute y = 0 in equation (1):
0 - y1 = (-x1/(2a))(x - x1)
x - x1 = 0
x = x1
Therefore, the x-coordinate of point R is x2 = x1.
7. Finding the length of AR: The length of AR is the difference between the x-coordinates of points A and R, which is:
AR = x2 - 0 = x1
Since x1 represents the x-coordinate of point P, which is on the parabola, it is equal to the focal distance of the point P. Therefore, the length of AR is equal to twice the focal distance of point P.
Hence, the correct answer is option 'C'.
1. Equation of the Parabola: The given equation y^2 = 4ax represents a parabola with vertex A at the origin (0,0) and the focus at point F(a,0). The axis of symmetry is the x-axis.
2. Chord PQ: The line segment PQ is a chord of the parabola passing through a point P(x1, y1). Since PQ is a normal chord, it is perpendicular to the axis of symmetry.
3. Line parallel to AQ: Through point P, a line is drawn parallel to AQ (the axis of symmetry). Let's call the point of intersection of this line with the x-axis as R(x2, 0).
We need to find the length of AR.
4. Finding the equation of line PQ: Since PQ is perpendicular to the axis of symmetry, its slope is the negative inverse of the slope of AQ. The slope of AQ can be found using the derivative of the parabola equation:
dy/dx = 2a/x
At point P(x1, y1), the slope of the tangent line to the parabola is dy/dx = 2a/x1. Therefore, the slope of PQ is -x1/(2a).
Using the slope-intercept form of a line, the equation of line PQ is:
y - y1 = (-x1/(2a))(x - x1) ... (1)
5. Finding the equation of line PR: Since PR is parallel to AQ, its slope is the same as that of AQ, which is 0 (since AQ is parallel to the x-axis).
Using the point-slope form of a line, the equation of line PR is:
y - 0 = 0(x - x2)
y = 0 ... (2)
6. Finding the point of intersection R: To find the x-coordinate of point R, we substitute y = 0 in equation (1):
0 - y1 = (-x1/(2a))(x - x1)
x - x1 = 0
x = x1
Therefore, the x-coordinate of point R is x2 = x1.
7. Finding the length of AR: The length of AR is the difference between the x-coordinates of points A and R, which is:
AR = x2 - 0 = x1
Since x1 represents the x-coordinate of point P, which is on the parabola, it is equal to the focal distance of the point P. Therefore, the length of AR is equal to twice the focal distance of point P.
Hence, the correct answer is option 'C'.
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PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer?
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PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer?.
PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer?.
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ample number of questions to practice PQ is a normal chord of the parabola y2= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR isa)Equal to the length of the latus rectumb)Equal to the focal distance of the point P.c)Equal to twice the focal distance of the point P.d)Equal to the distance of the point P from the directrixCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Class 11 tests.
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