Mathematics

If the straight line through the point $P(3,4)$ makes an angle $\dfrac{\pi}{6}$ with the $x-$axis and meets the line $12x+5y+10=0$ at $Q$, find the length of $PQ$.

SOLUTION
Any line through $P(3,4)$ making an angle  $\dfrac{\pi}{6}$ with $x-$axis is
$\dfrac { x-3 }{ \cos { { 30 }^{ o } } } =\dfrac { y-4 }{ \sin { { 30 }^{ o } } } =r$
Where $r$ represents the distance of any point on this line from the given point $P(3,4)$
Any point $Q$ on it is ${(r\sqrt {3}/2)+3,(r/2)+4}$ and $Q$ lies on $12x+5y+10=0$
$\therefore\quad 12\left[ \left( \dfrac { r\surd 3 }{ 2 } \right) +3 \right] +5\left( \dfrac { r }{ 2 } +4 \right) +10=0$
$\therefore\quad (12\sqrt {3}+5)r+132=0$
$\therefore\quad r=\dfrac { \left| -132 \right| }{ 12\surd \left( 3 \right) +5 } =\dfrac { 132 }{ 12\surd \left( 3 \right) +5 } .$

You're just one step away

Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86

Realted Questions

Q1 Subjective Medium
If two supplementary angles have equal measure, what is the measure of each angle?

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q2 One Word Medium
Find the  numbers of  pairs of vertically opposite angles in the figure.

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q3 Subjective Medium
Find the supplement of the following angle:
$123^\circ$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q4 Single Correct Medium
B lies on $\displaystyle \overline { AC } ,\overline { BD }$ bisects $\displaystyle \angle ABE$, and $\displaystyle \overline { BE }$ bisects $\displaystyle \angle CBD$ as shown in the above figure. Find the measure of $\displaystyle \angle DBE$.
• A. $\displaystyle { 90 }^{ \circ }$
• B. $\displaystyle { 45 }^{ \circ }$
• C. $\displaystyle { 30 }^{ \circ }$
• D. Cannot be determined from the given information
• E. $\displaystyle { 60 }^{ \circ }$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q5 Single Correct Medium
An angle whose measure is less than that of a right angle is ______.
• A. Acute
• B. Obtuse
• C. Right
• D. Straight

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020