Question 1 :
If R is any point on the x-axis and Q is any point on the y-axis and P is a variable point on RQ with RP = b, PQ = a. then find the equation of locus of P.
Solution :
QR = a + b
In triangle PAR sin θ = AP/PR sin θ = k/b ---(1) |
In triangle QBP cos θ = BP/QP cos θ = h/a ---(2) |
sin2θ + cos2θ = (k/b)2 + (h/a)2
1 = (k2/b2) + (h2/a2)
(h2/a2) + (k2/b2) = 1
Question 2 :
If the points P(6, 2) and Q(−2, 1) and R are the vertices of a ΔPQR and R is the point on the locus y = x2 − 3x + 4, then find the equation of the locus of centroid of ΔPQR
Solution :
Let R (a, b)
Centroid of the triangle PQR be (h, k)
P(6, 2) Q(-2, 1) R(a, b)
Centroid of the triangle = (x1+x2+x3)/3, (y1+y2+y3)/3
(h, k) = (6-2+a)/3, (2+1+b)/3
(h, k) = (4+a)/3, (3+b)/3
h = (4+a)/3 3h = 4 + a a = 3h - 4 |
k = (3+b)/3 3k = 3 + b b = 3k - 3 |
Since the point (a, b) lies on the curve y = x2 − 3x + 4
b = a2 − 3a + 4
3k - 3 = (3h - 4)2 − 3(3h - 4) + 4
3k - 3 = 9h2 + 16 + 24h + 9h + 12 + 4
9h2 + 33h - 3k + 32 + 3 = 0
9h2 + 33h - 3k + 35 = 0
Question 3 :
If Q is a point on the locus of x2 + y2 + 4x − 3y + 7 = 0, then find the equation of locus of P which divides segment OQ externally in the ratio 3:4, where O is origin.
Solution :
Let Q be (a, b) and P (h, k)
P is the point which divides the line segment joining the point OQ externally in the ratio 3 : 4
O (0, 0) Q (a, b)
= (mx2 - mx1)/(m - n), (my2 - my1)/(m - n)
= (3a - 0)/(3-4), (3b - 0)/(3-4)
(h, k) = (-3a, -3b)
-3a = h a = -h/3 |
-3b = k b = -k/3 |
Q is a point on the locus of x2 + y2 + 4x − 3y + 7 = 0
a2 + b2 + 4a − 3b + 7 = 0
By applying the values of "a" and "b" in the above equation, we get
(-h/3)2 + (-k/3)2 + 4(-h/3) - 3(-k/3) + 7 = 0
(h2/9) + (k2/9) - (4h/3) + k + 7 = 0
h2+ k2 - 12h + 9k + 63 = 0
Here h = x and k = y
x2+ y2 - 12x + 9y + 63 = 0
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