Science and Education
Co-ordinate Geometry
Q: The angle between the hyperbola xy = c2 and x2 – y2 = a2 is
(A) independent of c
(B) dependent on a
(C) always p/3
(D) none of these
$\large \frac{dy}{dx} = -\frac{y}{x}$
Differentiating x2 – y2 = a2 with respect to x, we get
$\large \frac{dy}{dx} = \frac{x}{y}$
Clearly the two curves always intersect orthogonally.
Hence (A) and (B) are the correct answers.
Q: The equation (x – α)2 + (y – β)2 = k(lx + my + n)2 represents
(A) a parabola for k = (l2 + m2)-1
(B) an ellipse for 0 < k < (l2 + m2)-1
(C) a hyperbola for k > (l2 + m2)-1
(D) a point circle for k = 0.
Sol. (x – α)2 + (y – β)2 = k(lx + my + n)2
$\large = k(l^2 + m^2)(\frac{lx + my + n}{\sqrt{l^2 + m^2}})^2 $
⇒ PS/PM = k( l2 + m2)
If k(l2 + m2) = 1 , ‘P’ lies on parabola
If k(l2 + m2) < 1 , ‘P’ lies on ellipse
If k(l2 + m2) >1 , ‘P’ lies on hyperbola
If k = 0, ‘P’ lies on a point circle.
Hence (A), (B), (C) and (D) are the correct answers.
Q: If a rectangular hyperbola (x – 1)(y – 2) = 4 cuts a circle x2 + y2 + 2gx + 2fy +c = 0 at points (3, 4), (5, 3), (2, 6) and (-1, 0), then the value of (g + f) is equal to
(A) 8
(B) -9
(C) -8
(D) 9
Sol. Let xiyi, i = 1, 2, 3, 4 be the points of intersection of hyperbola and circle, then
$\Large \frac{\Sigma x_i}{4} = \frac{-g+1}{2} $
g = -7/2
and $\Large \frac{\Sigma y_i}{4} = \frac{-f+1}{2} $
∴ f = -9/2
g + f = -8.
Hence (C) is the correct answer.
Q: The point of intersection of the curves whose parametric equations are x = t2 + 1, y = 2t and x = 2s, y = 2/s, is given by
(A) (1, -3)
(B) (2, 2)
(C) (-2, 4)
(D) (1, 2)
⇒ x-1 = y2/4
x = 2s , y = 2/s
⇒ xy = 4
For the point of intersection we have,
$\large \frac{4}{y}-1 = \frac{y^2}{4} $
⇒ y3 + 4y –16 = 0
⇒ y = 2 ⇒ x = 2
Hence (B) is the correct answer.