The x–intercept of the tangent at any arbitrary point of the curve is proportional to
The line = 1 touches the curve y = be–x/a at the point
If the subnormal at any point on y = a1 – n xn is of constant length, then the value of n is
If the tangent at P of the curve y2 = x3 intersects the curve again at Q and the straight lines OP, OQ make angles a, b with the x–axis, where `O' is the origin, then tan a/tan b has the value equal to
The length of the normal to the curve x = a(q + sin q), y = a (1 – cos q), at q = is
The beds of two rivers (within a certain region) are a parabola y = x2 and a straight line y = x – 2. These rivers are to be connected by a straight canal. The co-ordinates of the ends of the shortest canal can be
If the area of the triangle included between the axes and any tangent to the curve xn y = an is constant, then n is equal to
At (0, 0), the curve y2 = x3 + x2
For the curve x = t2 – 1, y = t2 – t, the tangent line is perpendicular to x-axis where
Given curve is x = t2 − 1,y = t2 − t
Derivating w.r.to t we get
d x/ dt =2t -------(1)
and dy / dt = 2t − 1 -------(2)
dividing (2) by (1) we get
dy / dx = (2t − 1) / 2t
Therefore, the slope of the tangent is (2t − 1) / 2t
Given that the tangent is perpendicular to x-axis. Therefore, tangent is parallel to y-axis.
We know that slope of y-axis is infinity and the slopes of the two parallel lines are equal.
Therefore, slope of the tangent is infinity.
Hence, (2t − 1) / 2t = 1/0
⟹ 2t = 0
⟹ t = 0