Straight Lines

 $\mathrm{m}\mathrm{a}\mathrm{x}\left\{n\right(A),n(B\left)\right\}\le n(A\cup B)\le n\left(U\right)$

$\Rightarrow 76\le 76+63-x\le 100$

$\Rightarrow -63\le -x\le -39$

$\Rightarrow 63\ge x\ge 39$

$L_1:4x+3y+2=0$  

$L_2:3x-4y-11=0$               

Since circle C touches the line L2 at Q intersection point L1 and L2 is (1, -2)

P lies of L₁

$\therefore P\left(x,-\frac{1}{3}\left(2+4x\right)\right)$               

Now,

PQ = 5 ? (x – 1)² + ${\left(\frac{4x+2}{3}-2\right)}^2=25$  

$\Rightarrow x=4,-2$             &n

C : 4x² + 4y² – 12x + 8y + k = 0

$?\left(1,-\frac{1}{3}\right)$               

Lies on or inside the C then

$4+\frac{4}{9}-12-\frac{8}{3}+k\le 0$

$\Rightarrow k\le \frac{92}{9}$               

Now, circle lies in 4^th quadrant centre

$\equiv \left(\frac{3}{2},-1\right)$               

$\therefore r<1\Rightarrow \sqrt[]{\frac{9}{4}+1-\frac{k}{4}}<1$      

28. Kindly consider the following

76. 

If point P be the junction between the lines

2x – 3y + 4 = 0 ______ (1)

3x + 4y – 5 = 0 ______ (2)

Solving (1) and (2) using 3 × (1) – 2 × (2) we get,

6x – 9y + 12 – (6x + 8y – 10) = 0

–17y + 22 = 0

y = $\frac{22}{17}$

And 2x = 3y– 4

=> 2x = 3 × $\frac{22}{17}$ – 4

x = $\frac{33}{17}$ – 2 = $\frac{33-34}{17}$ =&

Kindly go through the solution

Let A have the co-ordinate (x, o)

By laws of reflection

∠PAB = ∠ QAB = θ

And ∠ CAQ + θ = 90°

As normal is ⊥ to surface (x-axis)

⇒ ∠CAQ = 90° - θ

and ∠CAP = 90° + θ

73. The given eqⁿ of limes are.

9x + 6y – 7 = 0 ______ (1)

3x + 2y + b = 0 ______ (2)

Let P (x₀, y₀) be a point equidistant from (1) and (2) so

9x₀ + 6y - 7 = ± 3 (3x₀ + 2y₀ + 6)

When, 9x₀ + 6y₀ – 7 = 3 (3x₀ + 2y₀ + 6)

⇒ 9x₀ + 6y₀ - 7 = 9x₀ + 6y₀ + 18

⇒ - 7 = 18 which in not true

So, 9x₀ + 6y₀ -

72. The given eqⁿ of the lines are.

x + y ? 5 = 0 _______ (1)

3x ? 2y + 7 = 0 ______ (2)

Given, sum of perpendicular distance of P (x, y)  from the two lines is always 10 .

The above eqⁿ can be expressed as a linear combination Ax + By + C = 0 where A, B & C are constants representing a straight li

71.