1. A point lies on x-axis at a distance of 5 units from y-axis. What are its co-ordinates? What will be its co-ordinates, if it lies on y-axis at a distance of -5 units from x-axis?
Answers
Answer:
The given equation of the parabola i7s y2 - 4x - 4y = 0
⇒ y2 - 4y = 4x
⇒ y2 - 4y + 4 = 4x + 4, (Adding 4 on both sides)
⇒ (y - 2)2 = 4(x + 1) ……………………………….. (i)
Shifting the origin to the point (-1, 2) without rotating the axes and denoting the new coordinates with respect to these axes by X and Y, we have
x = X + (-1), y = Y + 2 ……………………………….. (ii)
Using these relations equation (i), reduces to
Y2 = 4X……………………………….. (iii)
This is of the form Y2 = 4aX. On comparing, we get 4a = 4 ⇒ a = 1.
The coordinates of the vertex with respect to new axes are (X = 0, Y = 0)
So, coordinates of the vertex with respect to old axes are (-1, 2), [Putting X= 0, Y = 0 in (ii)].
The coordinates of the focus with respect to new axes are (X = 1, Y = 0)
So, coordinates of the focus with respect to old axes are (0, 2), [Putting X= 1, Y = 0 in (ii)].
Equation of the directrix of the parabola with respect to new axes in X = -1
So, equation of the directrix of the parabola with respect to old asex is x = -2, [Putting X = -1, in (ii)].
Equation of the axis of the parabola with respect to new axes is Y = 0.
So, equation of axis with respect to old axes is y = 2, [Putting Y = 0, in (ii)].
The length of the latusrectum is 4 units.
2. Find the point on the parabola y2 = 12x at which the ordinate is double the abscissa.
Solution:
The given parabola is y2 = 12x.
Step-by-step explanation:
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Answer:
A circle of radius `2` units touches the co ordinate axes in the first quadrant. If the circle makes a complete rotation on the x-axis along the positive direction of the x-axis, then the equation of the circle in the new position