Length of Intercept Made by a Circle on a Straight Line

Q.

Let O be the centre of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{r}}^2$where $\mathrm{r}>\sqrt{\frac{5}{2}}$.

Q. Find the area bounded by x=at^2 , y=2at between the ordinates corresponding to t=1 and t=2

Q. Find the area of the region bounded by the straight line x-2y+8=0 and the hyperbole x²=8y.

Q.

If the lengths of the chords intercepted by the circle $x^2+y^2+2gx+2fy=0$ from the co-ordinate axes be 10 and 24 respectively, then the radius of the circle is

1. 17

2. 9

3. 14

4. 13

Q.

$\alpha ,\beta$ are the roots of the equation $x^2$ + bx + c = 0 and $\alpha +2,\beta +2$ are the roots of the equation $x^2$ + px + q=0. If the minimum value of the expression $x^2$ + bx + c is +2, find the minimum value of $x^2$ + px + q

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Q.

The length of the chord intercepted by the circle $x^2+y^2=r^2$ on the line $\frac{x}{a}+\frac{y}{b}=1$

1. $\sqrt{\frac{r^2(a^2+b^2)-a^2{b}^2}{a^2+b^2}}$

2. $2\sqrt{\frac{r^2(a^2+b^2)-a^2{b}^2}{a^2+b^2}}$

3. $2\frac{\sqrt{r^2(a^2+b^2)-a^2{b}^2}}{a^2+b^2}$

4. $2\frac{\sqrt{r^2(a^2+b^2)+a^2{b}^2}}{a^2+b^2}$

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. 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. 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. A line cuts the y-axis at a distance of $3$ units from the origin and makes an angle of ${30}^o$ with the positive direction of the x-axis. The equation of the line is

1. $y-\sqrt{3}x+3\sqrt{3}=0$
2. $x-\sqrt{3}y+3\sqrt{3}=0$
3. $x+\sqrt{3}y-3\sqrt{3}=0$
4. none of these

Q. find the equation of the circle which passes through the points (5, 0)&(1, 4)&whose centre lies on the line X+Y-3=0

Q. Solve this:
Q.9. Find the equation to the straight line which bisects, and is perpendicular to the straight line joining the points A (a, b) and B (a, b).

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