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Through the vertex O of the parabola, y2 = 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2 and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1 + cotq2 equals
- a)– 2tanf
- b)– 2tan(p - f)
- c)0
- d)2cotf
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are...
To find the value of cot(q1) * cot(q2), we need to determine the values of q1 and q2.
First, let's analyze the given information:
1. The equation of the parabola is y^2 = 4ax.
2. Chords OP and OQ are drawn through the vertex O of the parabola.
3. The circles on OP and OQ as diameters intersect at point R.
4. We need to find the angles q1 and q2 made with the axis by the tangent at P and Q on the parabola and by OR.
Now, let's find the coordinates of points P and Q:
1. The vertex of the parabola is O(0, 0).
2. The equation of the parabola is y^2 = 4ax, so the equation of the tangent at any point (x, y) on the parabola is y = (2a/x)x.
3. The tangent at P passes through point P, so we substitute the coordinates of P into the equation of the tangent:
y = (2a/x)x
0 = (2a/x)x
x = 0
Therefore, point P is P(0, 0).
4. Similarly, the tangent at Q passes through point Q, so we substitute the coordinates of Q into the equation of the tangent:
y = (2a/x)x
0 = (2a/x)x
x = 0
Therefore, point Q is Q(0, 0).
Now, let's find the coordinates of point R:
1. Since chords OP and OQ pass through the vertex O, they have the same y-coordinate as O.
Therefore, point R is R(x, 0).
2. The circles on OP and OQ as diameters intersect at point R. This means that the distance between O and R is equal to the sum of the radii of the circles.
The radius of the circle on OP is OP/2, and the radius of the circle on OQ is OQ/2.
The distance between O and R is (OP/2) + (OQ/2).
Since OP and OQ are chords passing through the vertex O, their lengths are equal to the distance between the vertex O and the point of intersection with the parabola.
Therefore, OP = OQ = 2a.
The distance between O and R is (OP/2) + (OQ/2) = (2a/2) + (2a/2) = 2a.
Therefore, the coordinates of point R are R(2a, 0).
Now, let's find the equations of the tangents at points P and Q:
1. The equation of the tangent at point P is y = (2a/x)x.
Substituting x = 0, we get y = (2a/0)0 = 0.
Therefore, the equation of the tangent at point P is y = 0.
2. The equation of the tangent at point Q is y = (2a/x)x.
Substituting x = 0, we get y = (2a/0)0 = 0.
Therefore, the equation of the tangent at point Q is y =
First, let's analyze the given information:
1. The equation of the parabola is y^2 = 4ax.
2. Chords OP and OQ are drawn through the vertex O of the parabola.
3. The circles on OP and OQ as diameters intersect at point R.
4. We need to find the angles q1 and q2 made with the axis by the tangent at P and Q on the parabola and by OR.
Now, let's find the coordinates of points P and Q:
1. The vertex of the parabola is O(0, 0).
2. The equation of the parabola is y^2 = 4ax, so the equation of the tangent at any point (x, y) on the parabola is y = (2a/x)x.
3. The tangent at P passes through point P, so we substitute the coordinates of P into the equation of the tangent:
y = (2a/x)x
0 = (2a/x)x
x = 0
Therefore, point P is P(0, 0).
4. Similarly, the tangent at Q passes through point Q, so we substitute the coordinates of Q into the equation of the tangent:
y = (2a/x)x
0 = (2a/x)x
x = 0
Therefore, point Q is Q(0, 0).
Now, let's find the coordinates of point R:
1. Since chords OP and OQ pass through the vertex O, they have the same y-coordinate as O.
Therefore, point R is R(x, 0).
2. The circles on OP and OQ as diameters intersect at point R. This means that the distance between O and R is equal to the sum of the radii of the circles.
The radius of the circle on OP is OP/2, and the radius of the circle on OQ is OQ/2.
The distance between O and R is (OP/2) + (OQ/2).
Since OP and OQ are chords passing through the vertex O, their lengths are equal to the distance between the vertex O and the point of intersection with the parabola.
Therefore, OP = OQ = 2a.
The distance between O and R is (OP/2) + (OQ/2) = (2a/2) + (2a/2) = 2a.
Therefore, the coordinates of point R are R(2a, 0).
Now, let's find the equations of the tangents at points P and Q:
1. The equation of the tangent at point P is y = (2a/x)x.
Substituting x = 0, we get y = (2a/0)0 = 0.
Therefore, the equation of the tangent at point P is y = 0.
2. The equation of the tangent at point Q is y = (2a/x)x.
Substituting x = 0, we get y = (2a/0)0 = 0.
Therefore, the equation of the tangent at point Q is y =
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Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect answer is option 'A'. Can you explain this answer?
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Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect 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 Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect 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 Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect answer is option 'A'. Can you explain this answer?.
Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect 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 Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect 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 Through the vertex O of the parabola, y2= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If q1, q2and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotq1+ cotq2equalsa)–2tanfb)–2tan(p - f)c)0d)2cotfCorrect answer is option 'A'. Can you explain this answer?.
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