Length of Intercept Made by a Circle on a Straight Line

Q.

Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and ${\left(2x\right)}^2+{\left(2y\right)}^2=31$. Then the square of the slope of the line $L$ is ___________.

Q.

If the common tangent to the parabolas, $y^2=4x$ and $x^2=4y$ also touches the circle, $x^2+y^2=c^2$, then $c$ is equal to:

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

2. $\frac{1}{4}$

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

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

Q.

The equation of tangents drawn from the origin to the circle $x^2+y^2-2rx-2hy+h^2=0$ are

1. $x=0,y=0$

2. $x=1,y=0$

3. $(h^2–r^2)x–2rhy=0,y=0$

4. $(h^2–r^2)x–2rhy=0,x=0$

Q.

If a tangent of slope $2$ of the ellipse $\left(\frac{x^2}{a^2}\right)+\left(\frac{y^2}{b^2}\right)=1$ is normal to the circle $x^2+y^2+4x+1=0$, then the maximum value of $ab$ is

1. $4$

2. $2$

3. $1$

4. None of these

Q. The equation of the circle which is touched by $y=x$, has its centre on the positive direction of the $x$-axis and cuts off a chord of length $2$ units along the line $\sqrt{3}y-x=0$, is

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

Q. Two chords are drawn from the point $P(h,k)$ on the circles $x^2+y^2-hx-ky=0.$ If the y-axis divides both the chords in the ratio $1:3,$ then which of the following is/are correct?

1. $9k^2>16h^2$
2. $k^2>48h^2$
3. $9k^2\ge 16h^2$
4. $4k^2=h^2$

Q. The locus of centre of a circle which passes through the origin and cuts off a length of $4$ units on the line $x=3$ is

1. $x^2+6y=13$
2. $y^2+6x=13$
3. $x^2+6y=0$
4. $y^2+6x=0$

Q. A circle $x^2+y^2+4x-2\sqrt{2}y+c=0$ is the director circle of circle $S_1$ and $S_1$ is the director circle of circle $S_2$ and so on. If the sum of radii of all these circles is $2$, then the value of $c=k\sqrt{2}$, where value of $k$ is

Q. The equation of circle passing through the origin and cutting off equal intercepts of $4$ units on the lines $xy=0$ can be

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

Q.

The circle $x^2+y^2=4$ cuts the line joining the points A(1, 0) and B(3, 4) in two points P and Q. Let $\frac{BP}{PA}=\alpha$ and

$\frac{BQ}{QA}=\beta$. Then $\alpha and\beta$ are roots of the quadratic equation

1. $3x^2+2x-21=0$

2. $3x^2+2x+21=0$

3. $2x^2+3x-21=0$

4. None of these

Q. The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$ by the lines, $x+y=n;n\in N$, where $N$ is the set of all the natural numbers, is :

1. $105$
2. $210$
3. $320$
4. $160$

Q. As shown in figure, three circles which have the same radius $r$, have centres at $(0,0),(1,1)$ and $(2,1)$. If they have a common tangent line, as shown, then the value of radius is

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

Q. Which of the following lines have the intercepts of equal lengths made by the circle $x^2+y^2-2x+4y=0$ is/are

1. $3x-y=0$
2. $x+3y=0$
3. $x+3y+10=0$
4. $3x-y-10=0$

Q. For any acute angle $\theta ,\tan (\theta -\frac{7\pi }{2}).\sec (\theta -\frac{7\pi }{2})=$

1. $\text{cosec}\theta {\sec }^2\theta$
2. $\sin \theta {\sec }^2\theta$
3. $\sin \theta {\cos }^2\theta$
4. $\text{cosec}\theta {\cos }^2\theta$

Q. For any acute angle $\theta$ select the trigonometric functions with a positive value.

1. $\sec \left(\frac{7\pi }{2}+\theta \right)$
2. $\mathrm{cosec}\left(\frac{5\pi }{2}+\theta \right)$
3. $\cos \left(\frac{5\pi }{2}-\theta \right)$
4. $\sin \left(\frac{3\pi }{2}-\theta \right)$

Q. The equation of the circle passing through the point (1, –2) and having its centre on the line 2x – y – 14 = 0 and touching the line 4x + 3y – 23 = 0

1. $x^2$ + $y^2$ + 8x + 12y + 27 = 0
2. $x^2$ + $y^2$ - 12y + 27 = 0
3. $x^2$ + $y^2$ - 8x + 12y - 27 = 0
4. $x^2$ + $y^2$ - 8x + 12y + 27 = 0

Q. Let $x^2+y^2-4x-2y-11=0$ be a circle. A pair of tangents from the point (4, 5) with a pair of radii form a quadrilateral of area ___.

Q. If a chord of the circle $x^2+y^2=8$ makes equal intercepts $a$ on the coordinate axes, then

1. $|a|<2$
2. $|a|\le 2\sqrt{2}$
3. $|a|<4$
4. $|a|<\sqrt{2}$

Q. The value of the expression $\tan 1°\times \tan 2°\times \tan 3°\times ...\times \tan 89°$ is :

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

Q. The value of $\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\cdots \tan {89}^{\circ }$ is