what is the required coordinate?

Question

Point P(−1,7) lies on the line 4x+3y=17. Then the coordinates of the points farthest from the line which are at a distance of 10 units from the point P are

Answer 1

Let A(x₁, y₁) & B (x₂, y₂) are required points.

\(\therefore \frac{y_1 - 7}{x_1 + 1} = \frac{-1}{\text{Slope of line } 4x + 3y = 17}\)

\(= \frac{-1}{\frac{-4}3} = \frac 34\)

⇒ \(y_1 -7 = \frac 34(x_1 + 1)\)

Also,

\((y, -7)^2 + (x_1 + 1)^2 = 100\)

⇒ \(\frac 9{16} (x_1 + 1)^2 + (x_1 + 1)^2 = 100\)

⇒ \(\frac{25}9 (x_1 + 1)^2 = 100\)

⇒ \((x_1 + 1)^2 = 100 \times \frac 9{25} = 36\)

⇒ \(x_1 + 1 = \pm 36\)

⇒ \(x_1 = 35 \, or\, -37.\)

⇒ \(y_1 = 34 \, or\, -20\)

Similarly,

\(\frac{y_2 - 7}{x_2+ 1} = \frac34 \)

⇒ \(y_2 -7 = \frac 34 (x_2 + 1)\)

And \((y_2 - 7)^2 + (x_2 + 1)^2 = 100\)

⇒ \((x_2 + 1)^2 = 36\)

⇒ \(x_2 = 35 \, or\, -37\)

\(\therefore y_2 = 34\, or\, -20\)

Hence, required points are (35, 34) & (-37, -20).