Q.1 PQ RS are the two perpendicular chords of the rectangular hyperbola xy = c^2.If C is the center of the rectangular hyperbola,then the product of the slopes of CP,CQ,CR&CSis equal to______
Q.2If PN is the perpendicular from a point P on the rectangular hyperbola to its asymptotes, the locus of the mid point of PN is _______ & how?
Q.1 To find the product of the slopes of the lines CP, CQ, CR, and CS, we need to first determine the equations of these lines.
Given that PQ and RS are perpendicular chords of the rectangular hyperbola xy = c^2, we can find the equations of these lines using the slope-intercept form of a line, y = mx + b.
Let's start by finding the equation of PQ. We know that PQ is perpendicular to RS. Since RS has a slope of 0 (as it is a horizontal line), the slope of PQ will be undefined (as it is a vertical line). Therefore, the equation of PQ will be x = a, where a is some constant.
Similarly, we can find the equation of RS. We know that RS is perpendicular to PQ, which means its slope will be the negative reciprocal of the slope of PQ. Since PQ is a vertical line, its slope is undefined. Therefore, the slope of RS will be 0. Hence, the equation of RS will be y = b, where b is some constant.
Next, let's find the equation of the line CP. The line CP connects the center C of the hyperbola to the point P. To find its slope, we need to determine the coordinates of C and P. However, the problem does not provide the specific coordinates of these points. Therefore, we cannot determine the equation of CP without additional information.
Similarly, we cannot determine the equations of the lines CQ, CR, and CS since the problem does not provide the coordinates of the points Q, R, and S. Thus, without more information, it is not possible to calculate the product of the slopes of CP, CQ, CR, and CS.
Q.2 To find the locus of the mid-point of PN, we need to determine the equation or description of the set of all points that satisfy this condition.
Let's start by understanding the given scenario. We have a point P on the rectangular hyperbola, and a perpendicular line PN is drawn from P to the asymptotes of the hyperbola. We want to find the locus of the midpoint of PN.
The midpoint of a line segment is the point that divides the line segment into two equal parts. In this case, the midpoint of PN will be equidistant from P and the closest asymptote.
Since the rectangular hyperbola has its asymptotes along the x and y axes, we can assume that the asymptotes are the lines y = 0 and x = 0.
To find the locus of the midpoint of PN, we can express the coordinates of the midpoint in terms of P, the coordinates of the asymptotes, and a parameter t. Let (x, y) be the coordinates of the midpoint of PN.
Using the Midpoint Formula, we can express the coordinates of the midpoint as follows:
x = (x_p + 0)/2 = x_p/2
y = (y_p + 0)/2 = y_p/2
Here, (x_p, y_p) are the coordinates of the point P on the hyperbola.
Now, since the midpoint of PN is equidistant from P and the asymptotes, we can use the Distance Formula to set up an equation.
The distance between (x, y) and (x_p, y_p) should be equal to the distance between (x, y) and the closest asymptote.
Using the Distance Formula, we get:
√((x - x_p)^2 + (y - y_p)^2) = d
where d is the distance between (x, y) and the closest asymptote, which is equal to half the distance between the asymptotes (since the midpoint is equidistant from P and both asymptotes).
The distance between the asymptotes is given by 2c, where c is a constant. Therefore, d = c.
Substituting this into our equation, we get:
√((x - x_p)^2 + (y - y_p)^2) = c
Squaring both sides of the equation and simplifying, we obtain:
(x - x_p)^2 + (y - y_p)^2 = c^2
This equation represents the locus of the midpoints of PN on the rectangular hyperbola xy = c^2.