Expression 'S1'

Q. Length of the tangent drawn from any point on the circle $x^2+y^2+2gx+2fy+c_1=0$ to the circle is $x^2+y^2+2gx+2fy+c=0$

1. $\sqrt{c_1-c}$
2. $\sqrt{c-c_1}$
3. $\sqrt{c_1+c}$
4. None of these

Q.

Let us get familiarized with some terms.

1) S: It is the equation of circle.

$S_1$: We get $S_1$ by substituting a point $x_1,y_1$ in the equation of the circle

$S_1=x_1^2+y_1^2+2g{x}_1+2f{y}_1+c$

2) T: We get T by replacing $x_2$ with $x{x}_1,y_2$ with $y{y}_1,x$ with $\frac{x+x_1}{2},y$ with $\frac{y+y_1}{2}$ and $xy$ wih $\frac{x{y}_1+y{x}_1}{2}.C$

remains the same

$T\equiv x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c$

These are really helpfull in finding the equation of tangent; chord of contact, pair of tangents from

appoint and determining if a point lies inside a circle.

If $S_1<0$, the point lies inside the circle

If $S_1=0$ the point lies on the circle and if $S_1>0$ the point lies outside the circle.

If you are comfortable and read this, click true to more to the question.

1. True

2. False

Q.

Which of the following point lie inside the circle $x^2+y^2-2x+3y-11=0$

1. (2, 3)

2. (1, 1)

3. (-1, 2)

4. (3, 2)

Q.

The point $(x_1,y_1)$ lies inside the circle $x^2+y^2+2gx+2fy+c=0$, if $x_1^2+y_1^2+2g{x}_1+2f{g}_1+c$ is less than zero

1. True

2. False

Q. Middle terms in expansion of $(\frac{a}{x}+bx{)}^{12}$ are $T_6$ and $T_7.$

1. True
2. False

Q. The middle term in the expansion of $(1+x{)}^{2n}$

1. $\frac{\mathrm{1.3.5}\cdots (2n-1)}{n!}{x}^n$
2. $\frac{\mathrm{2.4.6}\cdots 2n}{n!}{x}^n$
3. $\frac{\mathrm{1.3.5}\cdots (5n-1)}{n!}{x}^n$
4. $\frac{\mathrm{1.3.5}\cdots (2n-1)}{n!}{2}^n{x}^n$