Find the equation of the tangent line to the curve y =x² -2x + 7 which is (i) parallel to the line 2x-y +9 =0 (ii) perpendicular to the line 5y - 15x = 13.

Find the equation of the tangent to the curve $\begin{array}{r c l}{y}&{=}&{{\sqrt{3x-2}}}\end{array}$ which is parallel to the line 4x – 2y + 5 = 0.

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1.

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Find the equations of tangents to the curve 3x² – y² = 8, which passes through the point $\left({\frac{4}{3}},\,0\right).$

Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cos θ is at a constant distance from the origin.

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

Find the value of p for which the curves x² = 9p(9 – y) and x² = p(y + 1) cut each other at right angles.

Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ = ${\frac{\pi}{4}}\,.$