Differential Equations
Geometrical Applications of Differential Equations
Grade 12
Question:
<p>Let \(C\) be a curve such that the normal at any point \(P\) on it meets \(x\)-axis and \(y\)-axis at \(A\) and \(Y\) respectively. If \(BP:PA = 1:2\) (internally) and the curve passes through the point \((0, 4)\) then which of the following alternative(s) is/are correct?</p>
<p>(a) The curve passes through \(\left(\sqrt{10}, -6\right)\)</p>
<p>(b) The equation of tangent at \(\left(4, 4\sqrt{3}\right)\) is \(2x + \sqrt{3}y = 20\)</p>
<p>(c) The differential equation for the curve is \(yy'' + 2x = 0\)</p>
<p>(d) The curve represents a hyperbola</p>
Step-by-Step Solution
Key Concept: The normal at point P(x,y) has slope -1/dy/dx. Using the section formula with BP:PA = 1:2 (internally), we can express coordinates of A and B on the axes, then apply the collinearity condition to derive a differential equation relating x and y.
<p><strong>Step 1: Set up the normal line equation</strong></p><p>At point P(x,y) on curve C, let dy/dx = m. The normal has slope -1/m and equation: Y - y = (-1/m)(X - x)</p><p><strong>Step 2: Find intercepts A and B</strong></p><p>Point A (on x-axis, Y=0): 0 - y = (-1/m)(X - x) ⟹ X = x + my. So A = (x + my, 0)</p><p>Point B (on y-axis, X=0): Y - y = (-1/m)(0 - x) ⟹ Y = y + x/m. So B = (0, y + x/m)</p><p><strong>Step 3: Apply section formula with BP:PA = 1:2</strong></p><p>P divides BA internally in ratio 1:2 (from B to A):</p><p>x = (1·(x+my) + 2·0)/(1+2) = (x+my)/3 ⟹ 3x = x + my ⟹ <strong>my = 2x</strong></p><p>y = (1·0 + 2·(y + x/m))/(1+2) = (2y + 2x/m)/3 ⟹ 3y = 2y + 2x/m ⟹ <strong>y = 2x/m</strong></p><p><strong>Step 4: Solve the differential equation</strong></p><p>From my = 2x and y = 2x/m: Multiply to get my · y = 2x · 2x/m ⟹ m²y² = 4x²</p><p>Also from my = 2x: m = 2x/y, so (dy/dx) = 2x/y</p><p>This gives: y dy = 2x dx</p><p>Integrating: y²/2 = x² + C</p><p><strong>Step 5: Apply initial condition</strong></p><p>Curve passes through (0,4): 16/2 = 0 + C ⟹ C = 8</p><p>Therefore: y² = 2x² + 16 or <strong>y² - 2x² = 16</strong></p><p>∴ Answer: AB</p>
Correct Answer: AB