3D Geometry
Planes and Projections
Grade 12
Question:
<p>Through a point \(P(h, k, l)\) a plane is drawn at right angles to OP to meet the coordinate axes in A, B and C. If \(OP = p\), \(A_{xy}\) is area of projection of \(\triangle ABC\) on xy-plane, \(A_{yz}\) is area of projection of \(\triangle ABC\) on yz-plane, then \(\frac{A_{xy}}{A_{yz}}\)</p>
<p>(a) \(\Delta = \frac{p^5}{hkl}\)</p>
<p>(b) \(\Delta = \frac{p^5}{2hkl}\)</p>
<p>(c) \(A_{xy} = \frac{l}{h}\)</p>
<p>(d) \(A_{xy} = \frac{l}{h}A_{yz}\)</p>
Step-by-Step Solution
Key Concept: Use properties of projections of triangles on coordinate planes and the perpendicularity condition with OP.
The plane perpendicular to OP through P(h,k,l) has equation \(hx + ky + lz = h^2 + k^2 + l^2\). The projections of the triangle on coordinate planes are related by the direction cosines. The ratio \(\frac{A_{xy}}{A_{yz}} = \frac{l}{h}\).
Correct Answer: D