First slide

Equation of line in Straight lines

Question

Let $A (0, 0), B (5, 0), C (5, 3)$ and $D (0, 3)$ be the vertices of rectangle ABCD, If P is a variable point lying inside the rectangle ABCD and d (P, L) denotes perpendicular distance of point P from line L

Moderate

Question

If $d (P, A B)$ $\leq \min {d (P, B C), d (P, C D), d (P, A D)}$ then area of the region in which P lies is (in sq.units)

A

$\frac{17}{4}$
B

$\frac{19}{4}$
C

$\frac{21}{4}$
D

$\frac{23}{4}$

Solution

$d (P, A B) \leq d (P, B C)$
$d (P, A B) \leq d (P, C D)$
$d (P, A B) \leq d (P, A D)$
$\Rightarrow$ P lies in region as shown
Area $= \frac{1}{2} \times \frac{3}{2} \times 7 = \frac{21}{4}$

Question

If $d (P, A B)$ $\geq \max [d (P, B C), d (P, C D), d (P, A D)]$ then the area of the region in which P lies is (in sq.units)

A

1
B

$\frac{1}{2}$
C

$\frac{3}{4}$
D

$\frac{1}{4}$

Solution

$d (P, A B) \geq d (P, B C)$
$d (P, A B) \geq d (P, C D)$
$d (P, A B) \geq d (P, A D)$
$\Rightarrow$ P lies in region as shown
Area= $= \frac{1}{2} \times 1 \times \frac{1}{2} = \frac{1}{4}$

Question

If ${(d (P, A B) - \frac{3}{2})}^{2} + {(d (P, A D))}^{2} \geq 1$ , then area of region in which P lies is (in sq.units)

A

$15 - 2 π$
B

$10 - \frac{π}{2}$
C

$15 - π$
D

$15 - \frac{π}{2}$

Solution

$x^{2} + {(y - \frac{3}{2})}^{2} \geq 1$
P lies in region as shown
Area $= 5 \times 3 - \frac{π {(1)}^{2}}{2} = 15 - \frac{π}{2}$

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