The equation of tangent to the ellipse x² + 3y² = 3 which is r to line 4y = x – 5 is-

The equation of the tangents to the ellipse 4x² + 3y² = 5 which are parallel to the line y = 3x + 7 are

parallel lines have exactly equal slopes and perpendicular lines have the product of their slopes =-1. from this, we can find out the slope of the tangent of the ellipse.

The ellipse + = 1 and the straight line y = mx + c intersect in real points only if-

since the curves intersect at real points only, D should be greater than 0

The line x cos  + y sin  = p will be a tangent to the conic + = 1, if-

the tangent touches the curve at a single point and hence, has a single real solution to the equation.

Find the equations of tangents to the ellipse 9x² + 16y² = 144 which pass through the point (2,3).

equation of tangent of an ellipse is given by :
y= m+ √(a²m²+b²)
this is used to find the values of m.

Find the equation of the tangent to the ellipse x² + 2y² = 4 at the points where ordinate is 1.

the standard form of equation of tangents is obtained by substituting the values of coordinates into the equation of the curve

If + = touches the ellipse + = 1, then its eccentric angle θ is equal to-

the standard equation of an ellipse is obtained by substituting the parametric point on the curve equation.

The position of the point (4,– 3) with respect to the ellipse 2x² + 5y² = 20 is-

the position of a point with respect to a curve can be calculated by substituting the point and finding the resultant value
if it is less than zero,  then it lies inside the curve
equal to zero  then lies on the curve
greater than zero then lies outside the curve

The parametric representation of a point on the ellipse whose foci are (– 1, 0) and (7,0) and eccentricity 1/2 is-

the parametric form of a point on the ellipse gives us the coordinates of any point on the ellipse for a given angle.

Let P be a variable point on the ellipse + =1 with foci S and S'. If A be the area of triangle pss', then maximum value of A is–

the maxima or minima of a function is calculated by finding out the critical points of the function and then substituting the value of the critical point in the function.

If S and S' are two foci of an ellipse + = 1 (a < b) and P (x₁, y₁) a point on it, then SP + S' P is equal to-

the length of major axis of an ellipse is 2a for a horizontal ellipse and 2b for a vertical ellipse. a horizontal ellipse is formed when a>b and a vertical ellipse is formed when b>a. the major axis lies along x axis when a>b and along y axis when b>a.

The eccentricity of the ellipse represented by the equation 25x² + 16y² – 150x – 175 = 0 is -

the given equation has to be converted into the whole square form for x and y so that the standard form of ellipse can be observed. from the standard form, the eccentricity can be calculated by the values of a and b.

The foci of the ellipse,25 (x + 1)² + 9 (y + 2)² = 225, are at-

the ellipse has its vertex at (-1,-2). The terms x+1 and y+2 can be replaced with X and Y for better understanding so that it gets converted into the standard form.

The equation of the ellipse whose one of the vertices is (0, 7) and the corresponding directrix is y = 12, is-

this form of ellipse is a vertical one,i.e., the y axis is the major axis and the x axis is the minor axis.

The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is-

the focii are located at (ae,0) and (-ae,0). the distance between them is (ae-(-ae))= 2ae
similarly, the directrices are at a/e and -a/e .

Ionization energies of five elements in kcal/mol are given below:

The element having most stable oxidation state +2 is ?