Chord of Contact: Hyperbola

Q. If x = 9 is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is

1. $9x^2-8y^2+18x-9=0$
2. $9x^2-8y^2-18x+9=0$
3. $9x^2-8y^2-18x-9=0$
4. $9x^2-8y^2+18x+9=0$

Q. If the locus of the midpoint of contact of tangent drawn to the parabola $y^2=8x$ and foot of perpendicular drawn from its focus to the tangents is a conic then length of latus rectum of this conic is

1. $\frac{9}{4}$
2. $9$
3. $18$
4. $\frac{9}{2}$

Q. Let chords of the circle $x^2+y^2=a^2$ touch the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Then their middle points lie on the curve

1. $(x^2+y^2{)}^2=a^2{x}^2-b^2{y}^2$
2. $(x^2+y^2)=a^2{x}^2-b^2{y}^2$
3. $(x^2-y^2{)}^2=a^2{x}^2-b^2{y}^2$
4. $(x^2-y^2)=a^2{x}^2-b^2{y}^2$

Q. If the tangent to the parabola $y^2=4ax$ intersect the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $P$ and $Q$ and the locus of the point of intersection of the tangents at $P$ and $Q$ is $y^{\alpha }=-\frac{b^{\beta }}{a^{\gamma }}x$, then $\alpha +\beta +\gamma$ is

Q.

If the function $f\left(x\right)=x^3-6a{x}^2+5x$ satisfies the conditions of Lagranges mean value theorem for the interval $\left[1,2\right]$ and the tangent to the curve $y=f\left(x\right)$ at $x=\frac{7}{4}$ is parallel to the chord that curves the points of intersection of the curve with the ordinates $x=1$ and $x=2$. Then the value of $a$ is:

1. $\frac{35}{16}$

2. $\frac{35}{48}$

3. $\frac{7}{6}$

4. $\frac{5}{16}$

Q. If the chords of contact of tangents from points $(x_1,y_1)$ and $(x_2,y_2)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles such that $\frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^m}{b^n}$ where $m,n$ are positive integers, then the value of ${\left(\frac{m+n}{4}\right)}^{10}$ is

Q. The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ forms a triangle of constant area with the coordinate axes, is

1. A straight line
2. A hyperbola
3. An ellipse
4. A circle

Q. Let chords of the circle $x^2+y^2=a^2$ touch the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Then their middle points lie on the curve

1. $(x^2+y^2{)}^2=a^2{x}^2-b^2{y}^2$
2. $(x^2+y^2)=a^2{x}^2-b^2{y}^2$
3. $(x^2-y^2{)}^2=a^2{x}^2-b^2{y}^2$
4. $(x^2-y^2)=a^2{x}^2-b^2{y}^2$

Q. The locus of the points of intersection of the perpendicular tangents drawn to the hyperbola $\frac{x^2}{12}-\frac{y^2}{8}=1$ is the curve $S=0$. Again, the locus of the points of intersection of the perpendicular tangents drawn to the curve $S=0$ is another curve $S^{\prime }=0$. Then which of the following is TRUE?

1. $S=0$ and $S^{\prime }=0$ intersect in exactly $4$ points.
2. Tangent to $S^{\prime }=0$ intersect $S=0$ in exactly $2$ points.
3. Area bounded by the curve $S=0$ is $20\pi$ sq.units.
4. Area bounded by the curve $S^{\prime }=0$ is $8\pi$ sq. units

Q. Show that the line $x-y+4=0$ is a tangent to the ellipse $x^2+3y^2=12$. Find the point of contact

Q.

If the equations of four circles are $(x\pm 4{)}^2+(y\pm 4{)}^2=4^2$ then the radius of the smallest circle touching all the four circles is

1. 
2. 
3. 
4. 

Q. From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$ . Then, area cut-off by the chord of contact on the asymptotes is equal to

1. ab sq unit
2. a/2 sq unit
3. 2ab sq unit
4. 4ab sq unit

Q. The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ forms a triangle of constant area with the coordinate axes, is

1. A straight line
2. A hyperbola
3. An ellipse
4. A circle

Q. If chord contact of the tangents drawn from the point $(\alpha ,\beta )$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ touches the circle $x^2+y^2=c^2,$ THEN THE LOCUS OF THE POINT $(\alpha ,\beta )$

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{c^2}$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{c^4}$
3. $\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{c^2}$
4. none of these

Q. Angle between the common tangents of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{4}$

Q. From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$ . Then, area cut-off by the chord of contact on the asymptotes is equal to

1. a/2 sq unit
2. ab sq unit
3. 2ab sq unit
4. 4ab sq unit

Q.

The area bounded by the curve $y=x^2-1$, tangent to it at $\left(2,3\right)$ and Y-axis.

1. $\frac{2}{3}$

2. $\frac{4}{3}$

3. $\frac{8}{3}$

4. $1$

Q. If the solution of $\frac{16}{\pi }\frac{x}{x^2+y^2}dy=(\frac{y}{x^2+y^2}-1)dx$ satisfies$y\left(0\right)=1$ , then the value of $\frac{16}{\pi }y\left(\pi /4\right)$ is:

Q. Length of the shortest normal chord of the parabola $y^2=4ax$ is

1. $2a\sqrt{27}$
2. $a\sqrt{54}$
3. $9a$
4. none of these

Q. The locus of the middle points of chords of hyperbola $3x^2-2y^2+4x-6y=0$ parallel to y = 2x, is

1. 3x – 4y = 4
2. 3y – 4x + 4 = 0
3. 4x – 4y = 3
4. 3x – 4y = 2

Q. Find the locus of the middle points of all chords of the parabola $y^2=4ax$ which are drawn through the vertex.

1. $y^2=2ax$
2. $y^2=4ax$
3. $y^2=ax$
4. $2y^2=ax$

Q.

Find the equation to the chord of contact of tangents drawn from a point p(4, 3) to the hyperbola

$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. 3x - 4y = 12

2. 4x + 3y = 12

3. 3x + 4y = 12

4. 4x - 3y = 12

Q. A circle with centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is

1. $(\frac{p}{2},p)$
2. $(\frac{p}{2},2p)$
3. $(-\frac{p}{2},p)$
4. $(-\frac{p}{2},-p)$

Q. Locus of the point of intersection of perpendicular tangents drawn one each to the parabolas $y^2=4(x+1),y^2=8(x+2)$ is

1. $x+8=0$
2. $x+4=0$
3. $x+3=0$
4. $x+12=0$

Q. Prove that the locus of the poles of chords of a parabola which subtend a right angle at a fixed point (h, k) is
$a{x}^2-h{y}^2+(4a^2+2ah)x-2aky+a(h^2+k^2)=0.$

Q. The point of contact of the tangent to the parabola $y^2=9x$ which passes through the point $(4,10)$ and makes an angle $\theta$ with the axis of the parabola such that $\tan \theta >2$ is

1. $(\frac{4}{9},2)$
2. $(36,18)$
3. $(4,6)$
4. $(\frac{1}{4},\frac{3}{2})$

Q. A vector $\overrightarrow{r}$ has length $21$ and direction ratio's $2,-3,6$ the direction cosines of $\overrightarrow{r}$, given that $\overrightarrow{r}$ makes an obtuse angle with x-axis is given by

1. $\frac{2}{7},\frac{-3}{7},\frac{6}{7}$
2. $\frac{-2}{7},\frac{3}{7},\frac{-6}{7}$
3. $\frac{-2}{7},\frac{3}{7},\frac{6}{7}$
4. $1,0,0$

Q. From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$ . Then, area cut-off by the chord of contact on the asymptotes is equal to

1. a/2 sq unit
2. ab sq unit
3. 2ab sq unit
4. 4ab sq unit

Q.

Find the equation to the chord of contact of tangents drawn from a point p(4, 3) to the hyperbola

$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. 4x + 3y = 12

2. 4x - 3y = 12

3. 3x + 4y = 12

4. 3x - 4y = 12