Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c² to interest orthogonally.

Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.

Find the angle of intersection of the curves y² = 4ax and x² = 4by.

Show that the equation of normal at any point on the curve x = 3cos θ – cos³θ, y = 3sinθ – sin³θ is 4 (y cos³θ – x sin³θ) = 3 sin 4θ

The abscissa of the point on the curve 3y = 6x – 5x³, the normal at which passes through origin is ______.

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2 ______.

The tangent to the curve given by x = e^t . cost, y = e^t . sint at t = `pi/4` makes with x-axis an angle ______.

The equation of the normal to the curve y = sinx at (0, 0) is ______.

The point on the curve y² = x, where the tangent makes an angle of `pi/4` with x-axis is ______.

Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.

Find the condition that the curves 2x = y² and 2xy = k intersect orthogonally.

Prove that the curves xy = 4 and x² + y² = 8 touch each other.

Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes

Find the angle of intersection of the curves y = 4 – x² and y = x².

Prove that the curves y² = 4x and x² + y² – 6x + 1 = 0 touch each other at the point (1, 2)

Find the equation of the normal lines to the curve 3x² – y² = 8 which are parallel to the line x + 3y = 4.

At what points on the curve x² + y² – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?

Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e^– x/a at the point where the curve intersects the axis of y

If the straight line x cosα + y sinα = p touches the curve `x^2/"a"^2 + y^2/"b"^2` = 1, then prove that a² cos²α + b² sin²α = p².

The curve y = `x^(1/5)` has at (0, 0) ______.