For what value of $a$ do the curves $y = 1 - ax^2$ and $y = x^2$ intersect orthogonally?

The tangent to a non-linear curve $y = f(x)$ at any point $P(x, y)$ intersects the $x$-axis and $y$-axis at $A$ and $B$ respectively. If the normal to the curve $y = f(x)$ at $P$ intersects the $y$-axis at $C$ such that $AC = BC$,and $f(2) = 3$,then the equation of the curve is:

At which point on the curve $y^2 = x$ does the tangent make an angle of $45^{\circ}$ with the $x-$axis?

If the point $P(x_1, y_1)$ lying on the curve $y = x^2 - x + 1$ is the closest point to the line $y = x - 3$, then the perpendicular distance from $P$ to the line $3x + 4y - 2 = 0$ is

If the normal drawn at the point $P$ on the curve $y^2 = x^3 - x + 1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at $P$ is

If $y=4x-5$ is tangent to the curve $y^2=px^3+q$ at $(2,3)$,then