Properties of Tangent

Q. A tangent and a normal are drawn at the point $P(2,-4)$ on the parabola $y^2=8x$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a,b)$ is a point such that $AQBP$ is a square, then $2a+b$ is equal to

1. $-18$
2. $-12$
3. $-16$
4. $-20$

Q. If $P(-5,1)$ is one end of the focal chord $PQ$ of the parabola $x=y^2-8y+2,$ then the slope of the tangent at the other end is

Q. The locus of the foot of perpendiculars drawn from the vertex on a variable tangent to the parabola $y^2=4x$ is

1. $x(x^2+y^2)+x^2=0$
2. $y(x^2+y^2)+x^2=0$
3. $y(x^2+y^2)+y^2=0$
4. $x(x^2+y^2)+y^2=0$

Q. Let $x=my+c$ is normal to $x^2=4y$. If $k^2+mk+m=0$ is satisfies by only one real value of $k$, then value(s) of $c$ is/are

1. $0$
2. $-72$
3. $72$
4. $64$

Q. The ratio of the area of a triangle inscribed in a parabola to the area of the triangle formed by the tangents drawn at the vertices of the triangle is

Q. Let $x=my+c$ is normal to $x^2=4y$. If $k^2+mk+m=0$ is satisfies by only one real value of $k$, then value(s) of $c$ is/are

1. $0$
2. $-72$
3. $72$
4. $64$

Q. The length of the Subnormal at point $P\left(t\right)$ on the parabola $y^2=8x$ is equal to

Q. The slope of tangent of a curve, passing through $(3,4)$ at any point is the reciprocal of twice the ordinate of the point. Then the curve also passes through:

1. $(-4,3)$
2. $(-4,-3)$
3. $(12,5)$
4. $(12,-5)$

Q. If lines $x-y+2=0$ and $2x-y-2=0$ meet at a point $P,$ then equation of tangent drawn to the parabola $y^2=8x$ from the point $P$ is

1. $x-2y+8=0$
2. $x+y-16=0$
3. $3x-y-16=0$
4. $x-3y+16=0$

Q. Consider the parabola $y^2=8x$. Let ${\Delta }_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P(\frac{1}{2},2)$ on the parabola, and ${\Delta }_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\frac{{\Delta }_1}{{\Delta }_2}$ is

Q. If the straight line $(a-1)x-by+4=0$ is normal to the hyperbola $xy=1$ then which of the followings does not hold?

1. $a>1,b>0$
2. $a>1,b<0$
3. $a<1,b<0$
4. $a<1,b>0$

Q. If $P(-5,1)$ is one end of the focal chord $PQ$ of the parabola $x=y^2-8y+2,$ then the slope of the tangent at the other end is

Q. For a parabola, if $L_1:x=y+1$ is the axis of symmetry, $L_2:x+y=5$ is tangent at vertex and $L_3:y=4$ is a tangent at a point $P,$ then the equation of circumcircle of the triangle formed by the tangent and normal at point $P$ and axis of parabola is

1. $x^2+y^2+2y=31$
2. $x^2+y^2-2y=31$
3. $x^2+y^2+2x=31$
4. $x^2+y^2-2x=31$

Q. If $P(-5,1)$ is one end of the focal chord $PQ$ of the parabola $x=y^2-8y+2,$ then the slope of the tangent at the other end is

Q. If the area of region bounded by the parabola $y=x^2-4x+3$ and the straight lines touching it at the points with abscissae $x_1=1$ and $x_2=3\left(\text{in sq. units}\right)$ is $A,$ then the value of $3A$ is

Q. There are two perpendicular straight lines touching the parabola $y^2=4a(x+a)$ and $y^2=4b(x+b)$, then the point of intersection of these two lines lie on the line given by

1. $x+a+b=0$
2. $x-a-b=0$
3. $x-a+b=0$
4. $x+a-b=0$

Q. A vertical line passing through the point $(h,0)$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $P$ and $Q$.

1. $3$
2. $6$
3. $9$
4. $12$

Q. PQ is a double ordinate of a parabola. Find the locus of its points of trisection.

Q. If $P(-5,1)$ is one end of the focal chord $PQ$ of the parabola $x=y^2-8y+2,$ then the slope of the tangent at the other end is

Q. If the point $P(4,-2)$ is one end of the focal chord $PQ$ of the parabola $y^2=x$, then the slope of the tangent at $Q$ is