First slide

Equation of line in Straight lines

Question

The base of an isosceles triangle is equal to 4 units, the base angle is equal to $45^{0}$ . A straight line cuts the extension of the base at a point M at the angle $θ$ and bisects the lateral side of the triangle which is nearest to M.

Moderate

Question

The area of quadrilateral which the straight line cuts off from the given triangle is

A

$\frac{3 + \tan θ}{1 + \tan θ}$
B

$\frac{3 + 2 \tan θ}{1 + \tan θ}$
C

$\frac{3 + \tan θ}{1 - \tan θ}$
D

$\frac{3 + 5 \tan θ}{1 + t a n θ}$

Solution

Equation of line PM
$y - 1 = \tan θ (x - 1)$
Intersection point Q of AC and MP
$4 - x - 1 = \tan θ (x - 1)$
$\Rightarrow Q \equiv (\frac{3 + \tan θ}{1 + \tan θ}, \frac{1 + 3 \tan θ}{1 + \tan θ})$
Area of $Δ A P Q = \mod u l u s o f$
$\frac{1}{2} |\begin{matrix} 1 & 1 & 1 \\ 2 & 2 & 1 \\ \frac{3 + \tan θ}{1 + \tan θ} & \frac{1 + 3 \tan θ}{1 + \tan θ} & 1 \end{matrix}| = \frac{1 - \tan θ}{1 + \tan θ}$
Area of quadrilateral BPQC
$A = \frac{1}{2} \times 4 \times 2 - (\frac{1 - \tan θ}{1 + \tan θ})$
$= \frac{3 + 5 \tan θ}{1 + \tan θ}$

Question

The possible range of values in which area of quadrilateral which straight line cuts off from the given triangle lie in

A

$(\frac{5}{2}, \frac{7}{2})$
B

(4,5)
C

$(4, \frac{9}{2})$
D

(3,4)

Solution

$A = 5 - \frac{2}{1 + \tan θ}$
$θ = (0, \frac{π}{4}) \Rightarrow 1 + \tan θ \in (1, 2)$
$\Rightarrow A \in (3, 4)$

Question

The length of portion of straight line inside the triangle may lie in the range

A

(2,4)
B

$(\frac{3}{2}, \sqrt{3})$
C

$(\sqrt{2}, 2)$
D

$(\sqrt{2}, \sqrt{3})$

Solution

${(P Q)}^{2} = {(\frac{3 + \tan θ}{1 + \tan θ} - 1)}^{2} + {(\frac{1 + 3 \tan θ}{1 + \tan θ} - 1)}^{2}$
$\frac{4 (1 + \tan^{2} θ)}{1 + \tan^{2} θ + 2 \tan θ} = \frac{4}{1 + \sin 2 θ}$
$\Rightarrow {(P Q)}^{2} \in (2, 4) \Rightarrow P Q \in (\sqrt{2}, 2)$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App