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[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]?
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[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola int...
Let parabola=>
(y+k)^2 = 4a (x+h)
Pt. A (1,2)
(2+k)^2 = 4a (1+h) (1)
Pt. B (3,6)
(6+k)^2 = 4a (3+h) (2)
By subtracting (1) from (2), we get,
(6+k)^2 - (1+k)^2 = 4a [ (3+h) - (1+h)]
(8+2k) (4) = 4a(2)
8+2k = 2a
a=k+4 (3)
Eqn. of tangent at point A
(y+k) (2) = 2a (x+h+1) (4)
Eqn. of tangent at point B
(y+k) (6) = 2a (x+h+3) (5)
They intersect at (-1,1)
Putting value in eqn.(4)
2(1+k) = 2a(-1+h+1)
1+k = ah (6)
Putting value in eqn.(5)
6(1+k) = 2a(-1+h+3)
3(1+k) = a(2+h) (7)
From (6) and (7)
(1+k)/h = ( 3(1+k) / (2+h) )
(1+k) [2+h - 3h] = 0
k=-1,h=1
(1,-1) is center.
m (-1,1) to (1,-1) => -2/2 =-1
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[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola int...
Introduction: In this question, we are given two points on a parabola and the point of intersection of their tangents. We need to find the slope of the directrix of the parabola.
Understanding the problem: Before we start solving the problem, let us understand some basic concepts related to parabolas.
- A parabola is a set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
- The vertex of a parabola is the point where the parabola intersects its axis of symmetry.
- The axis of symmetry of a parabola is a line that passes through the vertex and is perpendicular to the directrix.
- The tangent at any point on a parabola is perpendicular to the axis of symmetry at that point.
Solution: Now, let us solve the problem step by step.
Step 1: Plot the given points and the point of intersection of their tangents on a graph.
Step 2: Draw the tangents at the given points and mark their slopes.
Step 3: Draw the line passing through the given point of intersection of the tangents and the vertex of the parabola. This line will be the axis of symmetry of the parabola.
Step 4: Since the tangents at the given points intersect at the given point, the axis of symmetry of the parabola must pass through this point as well.
Step 5: Find the equation of the line passing through the given point of intersection of the tangents and the vertex of the parabola. This equation will be of the form y = mx + c, where m is the slope of the line and c is a constant.
Step 6: Since the axis of symmetry of the parabola is perpendicular to the directrix, the slope of the directrix will be the negative reciprocal of the slope of the line found in Step 5.
Step 7: Find the negative reciprocal of the slope found in Step 5 to get the slope of the directrix.
Step 8: The required slope of the directrix is the negative reciprocal of the slope found in Step 5.
Conclusion: In this way, we can find the slope of the directrix of a parabola given two points on the parabola and the point of intersection of their tangents.
Understanding the problem: Before we start solving the problem, let us understand some basic concepts related to parabolas.
- A parabola is a set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
- The vertex of a parabola is the point where the parabola intersects its axis of symmetry.
- The axis of symmetry of a parabola is a line that passes through the vertex and is perpendicular to the directrix.
- The tangent at any point on a parabola is perpendicular to the axis of symmetry at that point.
Solution: Now, let us solve the problem step by step.
Step 1: Plot the given points and the point of intersection of their tangents on a graph.
Step 2: Draw the tangents at the given points and mark their slopes.
Step 3: Draw the line passing through the given point of intersection of the tangents and the vertex of the parabola. This line will be the axis of symmetry of the parabola.
Step 4: Since the tangents at the given points intersect at the given point, the axis of symmetry of the parabola must pass through this point as well.
Step 5: Find the equation of the line passing through the given point of intersection of the tangents and the vertex of the parabola. This equation will be of the form y = mx + c, where m is the slope of the line and c is a constant.
Step 6: Since the axis of symmetry of the parabola is perpendicular to the directrix, the slope of the directrix will be the negative reciprocal of the slope of the line found in Step 5.
Step 7: Find the negative reciprocal of the slope found in Step 5 to get the slope of the directrix.
Step 8: The required slope of the directrix is the negative reciprocal of the slope found in Step 5.
Conclusion: In this way, we can find the slope of the directrix of a parabola given two points on the parabola and the point of intersection of their tangents.
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[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]?
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[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about [" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for [" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]?.
[" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about [" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for [" If the tangents at two points "(1,2)" and "(3,6)" on a parabola intersect at the point "(-1,1)," then the "],[" slope of the directrix of the parabola is "]?.
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