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Numerade Educator



Problem 53 Hard Difficulty

Use a scalar projection to show that the distance from a point $ P_1 (x_1, y_1) $ to the line $ ax + by + c = 0 $ is
$$ \frac{\mid ax_1 + by_1 + c \mid}{\sqrt{a^2 + b^2 }} $$
Use this formula to find the distance from the point $ (-2, 3) $ to the line $ 3x - 4y + 5 = 0 $




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Video Transcript

the problem is used a Skylar projection To show that the distance from a point p one x one y want to the line. The ax, plus B y plus C is equal to zero is x one plus one plus seed value over squire it of a scrap raspy squire. You this family to find the distance from the point connected to three to the line. The reacts minus four y plus five is equal to zero. So first to look at is a graph here. If we choose some point zero zero one zero on the line X plus B plus C zero, then the scholar projection. Absolutely off the scale projection off the doctor. He's zero p one on two. And is it true? Thiss tens from Pew want to this line? You don't want to end? Is some wack to that are perpendicular to supply to the line X class B Y a seasick with zero Now, in this case, we can truth in as a bee. This water is a sovereign o to the line. A ex us b y plus C is equal to zero. Then we're half. So what? Repeat zero p Juan is he caught you Axel on minus like Cyril. Why one minus lives hero, then that this tense is culture to scan a projection off Dr P. Zero p one Hunt, you and Violet was this one. This is control E zero p Juan. Start in over magnitude. This is Cho Choo. I want absolute radio Axe one minus. Cyril, I have a plus. Why? One minus y zero times speed over your turf. A squire. Class B square. This's it. Could to absolute value. X one hams aid has x one minus Class B has why one minus eight hands. Like serial us. He taps y zero over a square US Peace Square. Since the point zero zero y zero is on the line. X Class B y last season zero. So we have a serial plus B Y zero us C is equal to zero. So a AK Cyril pass p Y zero is equal to negative C notice. It's equal to absolute value. X one plus one plus c over a tiff. A squared plus B squared. This is a foreigner here. Then we use this formula would find it. It's a distance from the point. Negative two three. Triple line reacts minus four y pass. Five. It's legal to zero, so the distance is equal to negative. Two times days is this in eighty six and three times negative force in negative. Twelve on class over. Tiff three. Squire plaice nectar, four square. The author is the teen over five.