Slope Point Form of a Line

Q. The number of distinct normals that can be drawn from $(-2,1)$ to the parabola $y^2-4x-2y-3=0$, is

1. $2$
2. $1$
3. $3$
4. $0$

Q.

If the circle $x^2+y^2+2ax+8y+16=0$
touches x-axis, then the value of a is

1. $\pm 4$
   

2. $\pm 8$

3. $\pm 1$

4. $\pm 16$
   

Q. The area bounded by the parabola $y^2=4ax$, its axis and two ordinates x=4, x=9 is

1. $4a^2$
   
2. $4a^2.4$
3. $4a^2(9-4)$
   
4. $\frac{152\sqrt{a}}{3}$

Q.

If $z=x+iy$, then area of the triangle whose vertices are points $z,iz,z+iz$ is

1. $\frac{1}{2}{\left|z\right|}^2$

2. $\frac{1}{4}{\left|z\right|}^2$

3. ${\left|z\right|}^2$

4. $\frac{3}{2}{\left|z\right|}^2$

Q. Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $y^2=4x.$ Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $△ACB$ is maximum. Then, the area (in sq. units) of $△ACB,$ is:

1. $31\frac{1}{4}$
2. $30\frac{1}{2}$
3. $32$
4. $31\frac{3}{4}$

Q. Through the vertex $O$ of the parabola $y^2=4ax$ a perpendicular is drawn to any tangent meeting it at $P$ and the parabola at $Q$. Then $OP\cdot OQ=$

1. $a^2$
2. $2a^2$
3. $3a^2$
4. $4a^2$

Q. Let $y=f\left(x\right)$ is a parabola of the form $y=x^2+ax+1$ and tangent to the parabola at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$.If it is known that no point of the parabola is below $x-$axis then the radius of circle when $a$ attains its maximum value in units is

1. $\frac{1}{\sqrt{5}}$
2. $1$
3. $\frac{1}{\sqrt{10}}$
4. $\sqrt{5}$

Q. The curve $x^2-y-\sqrt{5}x+1=0$ intersects $x-$axis at $A$ and $B.$ A circle is drawn passing through $A$ and $B.$ Then the length of the tangent drawn from the origin to the circle is

Q. Number of real normals to the parabola $y^2=16x,$ passing through $(4,0)$ is

1. $2$
2. $3$
3. $1$
4. $0$

Q.

If (-2, 5) is the centre of the circle which touches y-axis, then the equation of the circle is

1. x ²+y²-4x-10y+25=0

2. x ²+y²-4x-10y+4=0

3. x ²+y²+4x+10y+2=0

4. x ²+y²+4x-10y+25=0

Q. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^2=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $△PXQ$ is maximum. Then $4$ times this maximum area (in sq. units) is

Q. Through the vertex $O$ of the parabola $y^2=4ax$ a perpendicular is drawn to any tangent meeting it at $P$ and the parabola at $Q$. Then $OP\cdot OQ=$

1. $a^2$
2. $2a^2$
3. $3a^2$
4. $4a^2$

Q. If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissa are $x_1$ and $x_2,$ then the value of $m$ for which $x_1+x_2=0$ is

Q. Let $P$ be a point on the parabola, $x^2=4y$. If the distance of $P$ from the centre of the circle, $x^2+y^2+6x+8=0$ is minimum, then the equation of the tangent to the parabola at $P$, is :

1. $x-y+3=0$
2. $x+4y-2=0$
3. $x+y+1=0$
4. $x+2y=0$

Q. A line passing through the origin and the point $A(\sqrt{3},1)$ is rotated through an angle of ${15}^{\circ }$ about the origin in anti-clockwise direction. If the point $B$ be the new position of $A$, then the product of the co-ordinates of $B$ is

Q. If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissa are $x_1$ and $x_2,$ then the value of $m$ for which $x_1+x_2=0$ is

Q. Let P$(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.If the normal at the point P intersects the x axis at (9, 0) , then the eccentricity of the hyperbola is:

1. $\sqrt{\frac{3}{2}}$
2. $\sqrt{\frac{5}{2}}$
3. $\sqrt{2}$
4. $\sqrt{2}$

Q. Let $f\left(x\right)=x^4+3x+1,[-2,-1]$ then

1. f has exactly one zero in $[-2,-1]$
2. f has exactly two zeros in $[-2,-1]$
3. f has at least one zero in $[-2,-1]$
4. f has no zero in $[-2,-1]$

Q. Find the area bounded by the $X$ -axis the parabola $y=-x^2+4$ .

Q. Find the equation of circle whose diameter is the line joining the points $(0,-1)$ and $(2,3)$. Find also the intercept made by it on the axis of x .

Q. Let $y=f\left(x\right)$ is a parabola of the form $y=x^2+ax+1$ and tangent to the parabola at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$.If it is known that no point of the parabola is below $x-$axis then the radius of circle when $a$ attains its maximum value in units is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{1}{\sqrt{5}}$
3. $1$
4. $\sqrt{5}$

Q. Two equal parabolas with axes in opposite directions touch at a point O. From a point P on one of them are drawn tangents PQ and PQ' to the other. Prove that QQ' will touch the first parabola in P' where PP' is parallel to the common tangent at O.

Q.

the values of'a' for which both roots of the equation (a+1) x²-3ax+4a=0 are greater than unity

Q. The number of distinct normals that can be drawn from $(-2,1)$ to the parabola $y^2-4x-2y-3=0$, is

1. $1$
2. $2$
3. $3$
4. $0$

Q. The parametric equations of a curve are given by $x={\sec }^{2}t,y=\cot t.$ Tangent at $Pt=\frac{\pi }{4}$ meets the curve again at Q; then $PQ=?$

1. $\left(2\sqrt{5}\right).$
2. $\frac{\left(5\sqrt{5}\right)}{2}.$
3. $\frac{\left(3\sqrt{5}\right)}{2}.$
4. $\left(3\sqrt{5}\right).$

Q. Assertion :If $f\left(x\right)=(x-1)(x-2)(x-3),$ then area enclosed by $\left|f\left(x\right)\right|$ between the lines $x=2.2,x=2.8$ and $x-$axis is equal to ${\int }_{2.2}^{2.8}(x-1)(x-2)(x-3)dx$ Reason: $(x-1)(x-2)(x-3)\le 0$ for all $x\in [2.2,2.8]$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. Let $P$ be a point on the parabola, $x^2=4y$. If the distance of $P$ from the centre of the circle, $x^2+y^2+6x+8=0$ is minimum, then the equation of the tangent to the parabola at $P$, is :

1. $x+y+1=0$
2. $x+4y-2=0$
3. $x+2y=0$
4. $x-y+3=0$

Q. If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissa are $x_1$ and $x_2,$ then the value of $m$ for which $x_1+x_2=0$ is

Q. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^2=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $△PXQ$ is maximum. Then $4$ times this maximum area (in sq. units) is

Q. Let $A,B,C$ be three sets of complex numbers as defined below:
$A$ = {$z$ : Im $z\ge 1$ }
$B$ = {$z:\mid z-2-i\mid =3$}
$C$ = {$z:$Re$\left(\right(1-i\left)z\right)=\sqrt{2}$}
The number of elements in the set $A\cap B\cap C$ is

1. 0
2. 1
3. $\infty$
4. 2