Question
Download Solution PDFIf the foot of the perpendicular drawn from the point (0, k) to the line 3x - 4y - 5 = 0 is (3, 1), then what is the value of k?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
If two nonvertical lines are perpendicular, then the product of their slopes is −1.
The slope of a line passing through the distinct points (x1, y1) and (x2, y2) is \(\rm \frac{y_2-y_1}{x_2-x_1}\)
Calculation:
Slope of line passing through points (0, k) and (3, 1)
\(=\rm \frac{1-k}{3-0} \\=\frac{1-k}{3}\)
3x - 4y - 5 = 0
⇒4y = 3x - 5
⇒ y = \(\frac{3}{4}\rm x-\frac{5}{4}\)
So, the slope of line 3x - 4y - 5 = 0 is 3/4
Now since line OP and 3x - 4y - 5 = 0 are perepndicular
\(\rm \frac{1-k}{3}\times \frac{3}{4}=-1.....(\text {Product of slopes of perpendicular lines} )\\ \Rightarrow 1-k=-4\\ \Rightarrow k =5 \)
Hence, option (3) is correct.
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