PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Show that the following curve intersect orthogonally at the indicated point y² = 8x and 2x² +  y² = 10 at  \[\left( 1, 2\sqrt{2} \right)\] ?

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

Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?

Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?

Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?

If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve  \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a²cos^{2 \[\alpha\]} \[-\] b²sin² \[\alpha\] = p² ?

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?

Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?

If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?

Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?

Write the coordinates of the point on the curve y² = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis  ?

Write the angle made by the tangent to the curve x = e^t cos t, y = e^t sin t at \[t = \frac{\pi}{4}\] with the x-axis ?

Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?

Find the coordinates of the point on the curve y² = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?