If $f left parenthesis x right parenthesis equals not stretchy integral subscript blank superscript blank fraction numerator 2 blank s i n blank x minus blank s i n blank 2 x over denominator x to the power of 3 end exponent end fraction d x$ , where $x not equal to 0$ , then Limitx $rightwards arrow 0 f to the power of ´ end exponent left parenthesis x right parenthesis$ has the value

The equation of the directrix of the parabola $y to the power of 2 end exponent plus 4 y plus 4 x plus 2 equals 0 comma$ $i s colon$

The equation of the common tangent touching the circle $left parenthesis x minus 3 right parenthesis to the power of 2 end exponent plus y to the power of 2 end exponent equals 9$ and the parabola $y to the power of 2 end exponent equals 4 x$ above the $x$ ‐axis is:

If the line x‐1 =0 is the directrix of the parabola $y to the power of 2 end exponent minus k x plus 8 equals 0$ , then one of the values of $k$ is :

Assertion (A): Three normals are drawn from the point $P$ ’ with slopes $m subscript 1 end subscript comma m subscript 2 end subscript comma m subscript 3 end subscript$ to the parabola $y to the power of 2 end exponent equals 4 x$ If locus of ‘ $P$ ’ with $m subscript 1 end subscript m subscript 2 end subscript equals alpha$ is a part of the parabola itself then $alpha equals 2$
Reason (R): If normals at $left parenthesis x subscript 1 end subscript comma y subscript 1 end subscript right parenthesis comma left parenthesis x subscript 2 end subscript comma y subscript 2 end subscript right parenthesis$ and $left parenthesis y subscript 3 end subscript comma y subscript 3 end subscript right parenthesis$ are concurrent then $y subscript 1 end subscript plus y subscript 2 end subscript plus y subscript 3 end subscript equals 0$

ABCD and EFGC are squares and the curve $y equals k square root of x$ passes through the origin $D$ and the points $B$ and F The ratio $fraction numerator F G over denominator B C end fraction$ is:

Statement‐I :: With respect to a hyperbola $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction minus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ pependicular are drawn from a point (5, 0) on the lines $3 y plus-or-minus 4 x equals 0$ , then their feet lie on circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals 16.$
Statement‐II :: If from any foci of a hyperbola perpendicular are drawn on the asymptotes of the hyperbola then their feet lie on auxiliary circle.

A hyperbola, having the transverse axis of length $2 s i n$ , is confocal with the ellipse $3 x to the power of 2 end exponent plus 4 y to the power of 2 end exponent equals 12$ Then its equation is ‐

The latus rectum of the hyperbola $16 x to the power of 2 end exponent minus 9 y to the power of 2 end exponent equals 144$ is‐

Statement‐I :: If a point $open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses$ lies in the shaded region $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , show in the figure, then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 0$
Statement‐II :: $P left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ lies outside the hyperbola $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 1.$

Statement‐I The ellipse $fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ and $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ are congruent.
Statement‐II The ellipse $fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ and $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ have the same eccentricity.

For such questions, we should know properties of ellipse. We should know all the formulas related to ellipse. The axis which is larger is always the major axis.

Statement‐I The ellipse $fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ and $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ are congruent.
Statement‐II The ellipse $fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ and $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ have the same eccentricity.

For such questions, we should know properties of ellipse. We should know all the formulas related to ellipse. The axis which is larger is always the major axis.

The minimum area of triangle formed by tangent to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ and coordinate axes‐

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

Therefore, the eccentricity of the ellipse is $space space fraction numerator 1 over denominator square root of 2 end fraction$

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

Therefore, the eccentricity of the ellipse is $space space fraction numerator 1 over denominator square root of 2 end fraction$

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

Therefore, there are two values of c.

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

Therefore, there are two values of c.

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at