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NCERT · Class 11 · Mathematics · Straight LinesWhat is the perpendicular distance of the point $(3, -4)$ from the line $3x - 4y + 10 = 0$?

Q: What is the perpendicular distance of the point $(3, -4)$ from the line $3x - 4y + 10 = 0$?

The perpendicular distance $d$ from a point $(x1, y1)$ to the line $Ax + By + C = 0$ is given by $d = \frac{|Ax1 + By1 + C|}{\sqrt{A^2 + B^2}}$. Here $x1 = 3, y1 = -4, A = 3, B = -4, C = 10$. Substituting the values: $d = \frac{|3(3) - 4(-4) + 10|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 + 16 + 10|}{\sqrt{9 + 16}} = \frac{35}{\sqrt{25}} = \frac{35}{5} = 7$. Therefore, the correct option is C.

Step-by-Step Solution

The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. Here $x_1 = 3, y_1 = -4, A = 3, B = -4, C = 10$. Substituting the values: $d = \frac{|3(3) - 4(-4) + 10|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 + 16 + 10|}{\sqrt{9 + 16}} = \frac{35}{\sqrt{25}} = \frac{35}{5} = 7$. Therefore, the correct option is C.

Detailed Options Breakdown

Option : 3

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Straight Lines.

Option 1: 5

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Straight Lines.

Option 2: 7 (Correct Answer)

Correct choice. Refer to the step-by-step verified solution guidelines above for details.

Option 3: 9

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Straight Lines.

💡 Study Guide: This question tests core syllabus concepts from Straight Lines. For formulas, key summaries, and mock exam reference guides, read the full Straight Lines Revision Notes.

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