The point on the curve y² = x the tangent at which makes an angle of 45º with x-axis will be given by

The equation of the tangent to the parabola y = 2 + 4x –4x² with slope –4 is

equation of a straight line : y-y1=m(x-x1)
m is given, calculate x1 and y1 by differentiating the equation of parabola and equating it with the slope.

The equation of the tangent at vertex to the parabola 4y² + 6x = 8y + 7 is

if the slope of a line is infinite, then it is parallel to the y axis and the equation becomes x=a, where x is the x intercept.
Slope of the tangent : dy/dx 

The point where the line x + y = 1 touches the parabola y = x– x² , is-

the point of intersection of two curves can be found out by solving the equations of the two curves,

The equation of the tangent to the parabola y² = 4x at the point (1, 2) is-

Slope of tangent = dy/dx 
Equation of line: y-y1 = m(x-x1)

If a tangent to the parabola 4y² = x makes an angle of 60º with the x- axis, then its point of contact is-

point of contact = (a/m², 2a/m)
slope = tan ( angle made with the x axis)
this gives us the value of m
y^2 = x/4 = 4(1/16)x
a = 1/16

At which point the line x = my + touches the parabola x² =4ay

the point of contact is the point of intersection between the 2 curves, the parabola and the line. this can be obtained by solving the two equations.

For what value of k, the line 2y – x + k = 0 touches the parabola x² + 4y = 0-

when a quadratic equation has 2 equal roots, the D value is 0 . D= b^2 - 4ac.=0
this gives us the possible value(s) of k

An element, X has the following isotopic composition, the weighted average atomic mass of the naturally- occurring element 'X' is closest to :

The line x + my + n = 0 will touch the parabola y² = 4ax, if

a quadratic equation has equal roots if the D value is zero. this is the condition of the line touching the curve.

The straight line 2x + y –1= 0 meets the parabola y² = 4x in

when D> 0, two real and distinct roots
when D=0, real and equal roots
when D<0 imaginary roots.

The line y = mx + c may touch the parabola y² = 4a (x + a), if-

the solution of the two curves gives us the quadratic equation. D=0 gives us the required answer.

The straight line x + y = k touches the parabola y = x – x², if k =

since only one point of contact is present, D=0  gives the real root.

If the line x + y –1 = 0 touches the parabola y² = kx, then the value of k is-

we can also solve for the two lines and make the D=0 for the resultant quadratic equation.

If length of the two segments of focal chord to the parabola y² = 8ax are 2 and 4, then the value of a is-

the harmonic mean of the lengths (PS,QS) where S is the focus is equal to the half of latus rectum
the parametric coordinates of the ends of focal chords are (at^2, 2at) and (a/t^2,-2a/t)

If PSQ is the focal chord of the parabola y² = 8x such that SP = 6. Then the length SQ is-

the harmonic mean of the lengths (PS,QS) where S is the focus is equal to the half of latus rectum.